Is it a true proposition that composition of a projection onto any $2$-dimensional plane $\epsilon_2$ and rotation by angle $\dfrac{\pi}{2}$ in this plane gives always skew-symmetric matrix independently from dimension of a space where projection is executed ? In the other words:
if $ S = Proj_{\epsilon_2}Rot_{\epsilon_2}(\dfrac{\pi}{2})$, then $S=-S^T$.
How to prove it ?
It is also equivalent to the statement that matrix made from vectors of the basis ${e_1 ~ e_2~ e_3~ ... e_n} $ projected onto 2-dimensional plane and rotated by $\dfrac{\pi}{2}$ is skew-symmetric.
It is possible, I suppose, to use property of commutativity of projection and rotation proved earlier, but it might be only one step (probably decisive) towards skew-symmetry.
After 4 hours
It seems I have worked out the proof. $Proj_{\epsilon_2}$ can be written as $aa^T+bb^T$ where $a$ , $b$ are defining orthogonal unit vectors for 2-dimensional plane. We have
$ S^T = (Proj_{\epsilon_2}Rot_{\epsilon_2}(\dfrac{\pi}{2}))^T=(Rot_{\epsilon_2}(\dfrac{\pi}{2}))^TProj_{\epsilon_2}^T= (Rot_{\epsilon_2}(\dfrac{\pi}{2}))^{-1}(aa^T+bb^T)^T=(Rot_{\epsilon_2}(-\dfrac{\pi}{2}))(aa^T+bb^T)= -Rot_{\epsilon_2}(\dfrac{\pi}{2})Proj_{\epsilon_2}=-S$
As we see projection and rotation (simple rotation by ${\dfrac{\pi}{2}}$) give always skew-symmetric matrix.
Opened NEW question is whether any skew-symmetric matrix (N-dimensional) can be decomposed to a projection matrix (up to the scale) and rotation (maybe not always simple one) in N-dimensional space.
In 2-d and 3-d space it seems it is always possible.
After 2 months
I still don't have an answer for the question above:
is n-dimensional skew-symmetric matrix always decomposable into some rotation matrix and other, maybe projection or not ? (or maybe this property is limited only to 2 and 3 D?)