Let , A is a skew-symmetric matrix if and only if
Necessity: if A is a skew-symmetric matrix, then . Hence . Note , hence .
Sufficiency: if , then . I think we need the lemma that if A is a symmetric matrix, then if and only if . Since is a symmetric matrix, from we know . Hence A is skew-symmetric.
EDIT: The proof of sufficiency above is not correct. See here for a correct proof.
The below is a simpler proof.
, so ;
, so .