Rotation in Higher Dimensions In a world of three spatial dimensions plus time, every atom rotates around a line, the axis of rotation.  
In a world of $N$ spatial dimensions where $N$ is greater than 3, must every atom  rotate, and if so does it rotate around a line, a plane, or a subspace of smaller number of dimensions?
 A: *

*One may show that a general rotation $R\in SO(N)$ in $N\geq 2$ spatial dimensions can be composed
$$R ~=~ R_1\circ \ldots\circ R_{k} $$
of at most $k=[\frac{N}{2}] $ pairwise commuting rotations $$R_1,\ldots, R_{k}~\in~ SO(N)$$ that each leaves a co-dimension-2 subspace invariant (although not necessarily the same subspace). 

*More explicitly, given a rotation $R\in SO(N)$ there exists an orthonormal basis $(e_1, \ldots, e_N)$ [which may depend on $R$] such that the rotation $R$ is represented by a block-diagonal matrix of the form
$$ \begin{pmatrix} \cos\theta_1 & \sin\theta_1 & \cr 
-\sin\theta_1 & \cos\theta_1 & \cr 
&& \cos\theta_2 & \sin\theta_2 & \cr 
&&-\sin\theta_2 & \cos\theta_2 & \cr 
&&&& \ddots \cr
&&&&&\cos\theta_k & \sin\theta_k & \cr 
&&&&&-\sin\theta_k & \cos\theta_k & \cr 
&&&&&&&1\cr 
&&&&&&&&1\cr
&&&&&&&&& \ddots \cr
&&&&&&&&&&1\end{pmatrix}. $$

*The rotation $R$ itself is only guaranteed to leave invariant a dimension-1 subspace (=a line through the origin) if the space dimension $N$ is odd.
A: In 2d, a rotation matrix has the form 
$$
r(\theta)=\left(\begin{array}{cc}
\cos\theta&-\sin\theta\\
\sin\theta&\cos\theta\end{array}\right):=
\left(\begin{array}{cc}
c(\theta)&-s(\theta)\\
s(\theta)&c(\theta)\end{array}\right)
$$
and rotates vector in a plane.
In 3d a rotation matrix can be written as a product
$$
r_{12}(\psi)r_{13}(\theta)r_{12}(\varphi)
$$
where 
\begin{align}
r_{12}(\psi)&=\left(\begin{array}{ccc}
c(\psi)&-s(\psi)&0\\
s(\psi)&c(\psi)&0\\
0&0&1
\end{array}\right)\\
r_{13}(\theta)&=\left(\begin{array}{ccc}
c(\psi)&0&-s(\psi)\\
0&1&0\\
s(\psi)&0&c(\psi)
\end{array}\right)
\end{align}
leaving one axis invariant.  This axis can be identified by the row or column containing $0$s everywhere except for one entry.
In SO(4), one can write a rotation matrix as a sequence or $r_{ij}$ matrices.  $r_{12}$ would have the form
$$
r_{12}(\psi)=\left(\begin{array}{cccc}
c(\psi)&-s(\psi)&0&0\\
s(\psi)&c(\psi)&0&0\\
0&0&1&0\\
0&0&0&1
\end{array}\right)
$$
and so leaves a 2-dimensional subspace invariant.
An SO(4) matrix can be written in the factored form
$$
r_{34}(\beta_1)r_{23}(\beta_2)r_{12}(\beta_3)
r_{34}(\beta_4)r_{23}(\beta_5)r_{34}(\beta_6)
$$
by restricting to real values the entries of the $SU(4)$ matrix factored as as done here.  This is not by any means the only possible factorization.  
Obviously, an SO(5) rotation can be written in terms of matrices leaving a 3-dimensional subspace invariant etc.
