This is just a short remark about 3 x 3 skew-symmetric-matrix.
Theorem. If is nonzero and skew-symmetric, then two of its singular values are nonzero and equal, while the other one is zero.
Proof. If is nonzero and skew-symmetric, then is normal, so its singular values are the moduli of its eigenvalues. Moreover the eigenvalues are purely imaginary. Since the characteristic polynomial of is a univariate cubic with real coefficients, it has either one real root only or three real roots. If it has three real roots then they must be all zero. So , violating the assumption. Hence has one real root which is zero, and two nonzero purely imaginary roots which are conjugate to each other. Then these two imaginary roots have the same modulus. Therefore has two equal nonzero singular values while the other one is zero.
So how to describe all complex 3 by 3 matrices that have the properties mentioned in the above result? If having time we may add a conjecture later.