Are orthonormal matrices rotations? If we take a an orthonormal in $\mathbb{R}^{2\times 2}$, we know it has to be of the form 
$$A =\begin{pmatrix} a & b\\ -b & a\end{pmatrix}$$
such that $$a^2+b^2=1$$
(or the colums could be multiplied by $-1$, but this would make no difference for the following). Since it has these restrictions we can define $\vartheta$ such that $a=\cos\vartheta$, $b=\sin\vartheta$ and we see that $A$ is a rotation matrix. If the $(-1)$ multiplication is applied, the only difference is that we change the direction of the rotation, but is still a rotation.
I was wondering if this still holds for higher dimensions, i.e. if we have an orthonormal matrix in $\mathbb{R}^{n\times n}$ such that it can be written as
$$A = \sum_{i=1}^{d\leq n}R_i,$$
where $R_i$ are rotations around some axis. I am not necessarily interested in the deconstruction itself, only if there is something known about this and if I could read up on this somewhere. Intuitively I would say that this does not exist, or if it exists it will not be of the form as I suggested above, but this still makes it inconclusive for me.
 A: In $n$-dimensional case, it could be shown that such orthogonal matrices $\boldsymbol A$ are similar to a block-diagonal matrices 
$$
\begin{bmatrix}
\boldsymbol R_1 &  & & & & \\
& \boldsymbol R_2 &&&&\\
&& \boldsymbol R_3 &&& \\
&&& \ddots &&\\
&&&&\boldsymbol R_k &\\
&&&&& \boldsymbol I_{n-2k}
\end{bmatrix}
$$
when $\det(\boldsymbol A) =1$, or
$$\begin{bmatrix}
\boldsymbol R_1 &  & & & & \\
& \boldsymbol R_2 &&&&\\
&& \ddots &&& \\
&&&\boldsymbol R_k &&\\
&&&& \boldsymbol I_{n-2k-1} & \\
&&&&& -1 \end{bmatrix}
$$
when $\det(\boldsymbol A) =-1$. Here 
$$
\boldsymbol R_j =
\begin{bmatrix}
\cos(\varphi_j) & -\sin (\varphi_j)\\ \sin(\varphi_j) & \cos(\varphi_j)
\end{bmatrix}  \quad [j = 1, \ldots, k],
$$
and $\boldsymbol I_m$ is an $m \times m $ identity matrix.
Hence such a decomposition exists.
Reference: Linear Algebra Done Wrong. Sergei Treil [Available online]
A: Since an orthogonal matrix is normal, it is diagonalizable over $\mathbb C$. Since it is unitary, its eigenvalues have magnitude $1$. Since its characteristic polynomial is real, its eigenvalues come in complex conjugate pairs. If you order the eigenvalues such that the pairs are consecutive, the diagonal blocks
$$
\pmatrix{\mathrm e^{\mathrm i\phi}&0\\0&\mathrm e^{-\mathrm i\phi}}
$$
can be transformed to
$$
\pmatrix{\cos\phi&-\sin\phi\\\sin\phi&\cos\phi}\;.
$$
Thus, an orthogonal transformation can be written as the product (not sum) of reflections and rotations in planes. In three dimensions, specifying a plane of rotation and a rotation axis is equivalent, but only the specification by a plane generalizes to higher dimensions.
An eigenvector with eigenvalue $1$ is invariant under the transformation; an eigenvector with eigenvalue $-1$ is reflected by the transformation; and each pair of eigenvectors with complex conjugate eigenvalues spans a plane of rotation.
