What is the geometrical action of a skew-symmetric matrix on an arbitrary vector?
The rotation matrix is a skew-symmetric matrix when $\theta$ is some multiple of $\frac{\pi}{2}$. But it cannot be true that every skew-symmetric matrix represents a rotation?
Also, since the leading diagonal is zero, it cannot represent a scaling nor a shear. In fact, none of the standard transformation matrices on Wikipedia seem to fit the pattern of an arbitrary skew-symmetric matrix.
So can anything be said about the geometrical action of a skew-symmetric matrix on an arbitrary vector?