What is a skew-symmetric matrix?
Solution 1:
It is a square matrix $A$ satisfying $A^T=-A$---that is, its transpose is the negative of the original. If $B=A^T$, then $B_{ij}=-A_{ji}$.
A similar property for complex matrices is skew-Hermitian: $A^H=-A$. If $B=A^H$, then $B_{ij}=-\overline{A_{ji}}=-\Re(A_{ji})+\jmath\Im(A_{ji})$.
Note that these properties require the diagonal elements to be zero.