Can rotations in 4D be given an explicit matrix form? Rotation in 2D by an angle $t$ can be performed using $$R=\begin{pmatrix}\cos(t) &-\sin(t) \\ \sin(t) & \cos(t)\end{pmatrix}$$ matrix. But, if I want to rotate a point or vector in 4D, is there any rotation matrix in explicit form?
I have read rotation about planes in 4D (Rotating two planes while the other two planes remains constant), but I am interested in rotation around an axis. More specifically the Quaternion 4D rotation matrix.
Kindly help me out. 
 A: Rotations in 2-D space rotate around a point (rotation center) which remains invariant under the rotation. Rotations in 3-D space rotate around a line (rotation axis) which remains invariant under the rotation. In 4-D space there are two sorts of rotations: a) Single rotations around a plane which remains invariant under the rotation, and b) Double rotations around a point (rotation center), which remains invariant under the rotation. General double rotations have two mutually independent rotation angles. Isoclinic rotations are double rotations wherein both rotation angles have the same absolute value; they can, however, still differ in relative sign, giving rise to left-isoclinic and right-isoclinic rotations.
Quaternions are isoclinic rotation operators:
A unit quaternion $q = (q_0 + iq_1 + jq_2 + kq_3)$ can always be written as
$$q(\varphi)= \big(\cos(\varphi) + i u_1 \sin(\varphi) + j u_2 \sin(\varphi) + k u_3 \sin(\varphi)\big)$$
$u = (u_1, u_2, u_3)$ being a 3D unit vector, the axis of rotation; and
$\varphi$ being the rotation angle around this axis $u$ .
Pre-multiplication of a 4-D point $P$ with a unit quaternion $q(\varphi)$ does, however, not only cause a 3D-rotation of $P$ by the angle $\varphi$ around $u$ ("curl") but also a "inward-outward" rotation of $P$ by the angle $\varphi$ along the fourth dimension ("source"). Taking the negative rotation angle $q(-\varphi)$ reverses the sense of both, "curl" and "source". Post-multiplication reverses the sense of the "curl" but not that of the "source".
To obtain a pure 3D-rotation by an angle $\varphi$, one can combine a premultiplication by $q(\varphi/2)$ with a post-multiplication by $q(-\varphi/2)$. The "curl"-rotations will then add up to the full rotation angle, whereas the "source"-rotations will mutually cancel each other.
A: Right from the creation of the universe things have always seemed complicated except if simplified especially in rotating and expanding cosmos like earthlings are used too.
Plane Of Rotation
When the rotation of an object takes place about the z-axis in 3D for example, the plane of rotation is the xy/yx plane i.e. x,y coordinates transform as the object turns about the z-axis thus points on the z-axis are free/do not partake in the turning therefore the plane of rotation rather than the  axis that happens to be perpendicular to the plane is where rotation really happens
Hence for 3D(x,y,z):
xy/yx plane rotation => z axis is free
xz/zx plane rotation => y axis is free
yz/zy plane rotation => x axis is free

PS: Usually when humans (I am a Cyborg from 2050 on a quest to retrieve the timestone from the earth's core
) say they are rotating the object about the z-axis in 3D, the plane of rotation is xy/yx and the object's transformation would be idle about the free axis(the z-axis)


So for 4D(x,y,z,w):
xy/yx plane rotation => z axis & w axis are free
xz/zx plane rotation => y axis & w axis are free
yz/zy plane rotation => x axis & w axis are free
xw/wx plane rotation => y axis & z axis are free
yw/wy plane rotation => x axis & z axis are free
zw/wz plane rotation => x axis & y axis are free
$$Rotation Transforms$$
3D Rotation Matrix: anti-clockwise order($xy,yz,zx$)
z-axis rotation
$$Rxy = \begin{pmatrix}
cos(\theta)&-sin(\theta)&0\\
sin(\theta)&cos(\theta)&0\\
0&0&1\\
\end{pmatrix}$$
x-axis rotation
$$Ryz = \begin{pmatrix}
1&0&0\\
0&cos(\theta)&-sin(\theta)\\
0&sin(\theta)&cos(\theta)\\
\end{pmatrix}$$
y-axis rotation
$$Rzx = \begin{pmatrix}
cos(\theta)&0&sin(\theta)\\
0&1&0\\
-sin(\theta)&0&cos(\theta)\\
\end{pmatrix}$$
4D Rotation Matrix: no clock order
zw-axis rotation
$$Rxy = \left[\begin{matrix} 
  cos(\theta) & -sin(\theta) & 0 & 0 \\
  sin(\theta) & cos(\theta) & 0 & 0 \\
  0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 1 \end{matrix}\right]$$
yw-axis rotation
$$Rxz = \left[\begin{matrix} cos(\theta) & 0 & -sin(\theta) & 0 \\
 0 & 1 & 0 & 0 \\ 
 sin(\theta) & 0 & cos(\theta) & 0 \\ 
 0 & 0 & 0 & 1 \end{matrix}\right]$$
yz-axis rotation
$$Rxw = \left[\begin{matrix} cos(\theta) & 0 & 0 & -sin(\theta) \\
  0 & 1 & 0 & 0 \\
  0 & 0 & 1 & 0 \\
  sin(\theta) & 0 & 0 & cos(\theta) \end{matrix}\right]$$
xw-axis rotation
$$Ryz = \left[\begin{matrix} 1 & 0 & 0 & 0 \\
  0 & cos(\theta) & -sin(\theta) & 0 \\
  0 & sin(\theta) & cos(\theta) & 0 \\
  0 & 0 & 0 & 1 \end{matrix}\right]$$
xz-axis rotation
$$Ryw = \left[\begin{matrix} 1 & 0 & 0 & 0 \\
 0 & cos(\theta) & 0 & -sin(\theta) \\
 0 & 0 & 1 & 0 \\
 0 & sin(\theta) & 0 & cos(\theta) \end{matrix}\right]$$
xy-axis rotation
$$Rzw = \left[\begin{matrix} 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & cos(\theta) & -sin(\theta) \\
 0 & 0 & sin(\theta) & cos(\theta) \end{matrix}\right]$$
The reason why 4D Rotations can't have Ryzw,  Rxzw,  Rxyw,  Rxyz is because generally rotation is the circular movement about a fixed point/pivot and it requires at least/only two axis for transformation in our 2-biased-state-universe (including the imaginary multiverses too, given our scope hasn't travelled  ♧D and beyond).
so that in 2D to rotate a square about a pivot a 0D point is at least stationary/invariant
also in 3D to rotate a cube about a pivot for a specific plane a 1D line passing through the pivot is stationary/invariant
and finally in 4D to rotate a tessaract about a pivot for a specific plane a 2D rect containing the pivot is at least stationary/invariant e.t.c.
I guess the multiverses decided the trend must be 0D point, 1D line, 2D rect...
visit the
universe and the
multiverses
Preparing the JetPack Vortex Experience 2.0:



Note how the Rotation Transform Hash seems to portray rotation only as a 2-axes phenomenon (R## from  2D - 5D) and how its red hashes seem analogous to the invariant pivot element.
The fundamental property of rotations:
To simply put in human understandable terms "a point if truly rotated, maintains a constant distance to its turning point (Pivot)" i.e. rotating a shape is only possible with each point in the shape maintaining a constant distance to the pivot. E.g. When a 3D point is rotated along the z axis (as humans would have it) two coordinates (x,y) would transform hence Rxy denotes this because  x and y axis values change and together remain equidistant to the turning point.
With that said, if a point transforms in x,y,z values while remaining equidistant to a pivot should that not imply Rxyz is possible? Yes but what one discovers would be that Rxyz = Rxy . Rxz or Rxyz = Rxy . Ryz
i.e.
Rxyz = (z-axis and y-axis)rotation  or Rxyz = (z-axis and x-axis)rotation

This is the reason it is argued Rxyz isn't possible by humans (which I strongly  disagree with). The same applies to rotations in all other environments with extra dimensions. So to discredit what the chief Cyborg said earlier (a little bit) Rxyz is possible in a 4D environment and beyond because It's still a 2axes-biased-rotation-universe (You may interpret this to mean the angular velocity of Spiderman's web-flinging in the MCU may fail if he gets permitted into the DCEU (heard the Bat only permits heroes with capes outta the stargate). I had  a  tachyon-skype call a few days back with Aquaman from the mariana trench at the other side and he said the word on the street nowadays is that they can rotate themselves when flying with their stomachs inside out (there were rumours of some alien crustaceans flying and swimming inside their waters too, I mean how should any normal thing fly inside water?) but I suspect it's a 3axes-biased-rotation-universe given the quantum-belly experience is astonishingly different, with that said one could only imagine if DCEU rotation sucks or not but this doesn't mean one would love the MCU more, although their characters are so much cooler (don't blame me the DCEU Bat has always been a Joker (pun intended), I mean he's supposed to use his big ears to navigate while flying but all he does is open his eyes as wide as possible)).

A: To answer the question "I am interested in rotation around an axis".  
Rotation is an inherently $2$-D operation: rotate the vector $<x,y>$ by an angle $\theta$. So, strictly speaking, there is never an "axis of rotation". In fact what happens is that there is a rotation about a point in the plane of rotation; either the origin, or some other point. Everything is within the plane.
For rotations in $3$-D space and above - all you need to do is specify the $2$-D plane of rotation within the space.
If you want to talk about "axes of rotation" in a non-formal way, to help you understand things in $3$-D space, or spaces above that, then starting from $2$-D space upwards:  
In $2$-D space, there is no "axis of rotation".
In $3$-D space, there is a $1$-D "axis of rotation"; a line.
In $4$-D space, there is a $2$-D "axis of rotation"; a plane.
In $5$-D space, there is a $3$-D "axis of rotation"; a $3$-D space.
and so on ...
Following on from @JohnHughes ' answer, although one cannot write, in $n$-D space, where $n > 3$, that for rotation about "an axis" $L$, that for any vector $v$ in the "axis" $L$, $Rv = v$, and that for any vector $w$ orthogonal to the "axis" $L$, $Rw \cdot w = \cos \theta$, as there are more than one $1$-D axes orthogonal to the $2$-D surface (plane) (in which the rotation occurs); in such spaces, one can say:
Take $S$ as the $2$-D surface (plane) in which the rotation occurs,
Take $S'_{i=1..m}$ as all the $2$-D surfaces (where the two orthogonal axes of $S'_i$ are each orthogonal to $S$) - where $m = \frac{(n-2)!}{2!((n-2)-2)!}$
Then, 


*

*for any $v$ in $S$, $Rv \cdot v = \cos \theta$, and

*for any $w$ in $S'_i$, $Rv = v$
