High Dimensional Rotation Matrices As Product of In-Plane Rotations Lately I've been thinking a lot about how to find high-dimensional rotation matrices. In particular, can any rotation in $n$-dimensional space be represented as the product of $2$D plane rotations? I'm having a tough time finding this online. 
For example, in $4$D, could we just take rotations in the planes $XY$, $XZ$, $YZ$, $XW$, $YW$, $ZW$ and multiply them together? Obviously rotation matrix multiplication is noncommutative, so will any ordering of the multiplication result in an expression that could be used to represent any high-dimentional rotation? 
Thanks!
 A: Your conjecture is true.
The special orthogonal group $\text{SO}(n)$ is the set of all $n\times n$ matrices $R$ satisfying $R^T R=I$ and $\det(R)=1$. This is a compact lie group and its associated Lie algebra $\mathfrak{so}(n,\mathbb{R})$, which is the tangent space at the identity element, is equal to $\text{Skew}_n$, the vector space of $n\times n$ skew symmetric matrices over $\mathbb{R}$, that is real matrices satisfying $A^T=-A$. Since $\text{SO}(n)$ is a connected compact Lie group, the exponential map will be a surjective map from $\mathfrak{so}(n,\mathbb{R})$ to $\text{SO}(n)$. In other words, every element of $\text{SO}(n)$ can be written as $\exp(A)$ for a skew symmetric matrix $A$. 
The vector space $\text{Skew}_n$ has the basis $\mathcal{B}=\{\mathcal{S}_{ab}:\ b>a\geq 1\}$, where the matrix $\mathcal{S}_{ab}$ has a $1$ in the $ab^{th}$ position, and a $-1$ in the $ba^{th}$ position. For example, for $n=4$, $$\mathcal{S}_{12}=\left[\begin{array}{cccc}
0 & 1 & 0 & 0\\
-1 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{array}\right].$$
Notice that this basis has cardinality $n(n-1)/2$, and that $\exp(\mathcal{S}_{ab})$ will correspond to a rotation around an axis for each $a,b$. Thus $\mathcal{SO}(n)$ is an $n(n-1)/2$ parameter family generated by rotations in $n(n-1)/2$ different planes.
A: Yes. First off, given any transformation $T:V\to V$ of real vector spaces, we can decompose $V$ into a direct sum of invariant lines and planes. This can be deduced from either putting it into rational canonical form, or considering $V$ to be a $\mathbb{R}[X]$-module where $X$ acts as $T$ and then invoking the fundamental theorem of finitely-generated modules over PIDs. (I think of the latter as the route to proving rational canonical forms exist in the first place.) In any case, $T$ must have an invariant line or plane.
If $R$ is a rotation in an inner product space, and it has an invariant line or plane $S$, then $R$ restricts to rotations on $S$ and its orthogonal complement $S^\perp$ (can you prove this?). One can then proceed by induction to gather $R$ is a product of commuting orthogonal plane rotations.
I am myself interested in finding a more elementary, preferably geometric, argument for why the rotation should have an invariant line or plane, perhaps only utilizing basic facts of inner products, and also I asked a question about extending this result to one-parameter subgroups of rotations, or equivalently about extending the result to the Lie algebra $\mathfrak{so}(n)$.
