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?
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.
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.
