Prove/disprove: if A, a matrix nxn over field F is skew-symmetric then A congruents with a diagonal matrix.
I know that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. But is it true for skew-symmetric as well? I really have no idea what to do here... I know A consists of zeros on it's diagonal. it means it's nilpotent? if so, the eigenvalues are only zeros, I guess it means that its diagonal matrix should be all zeros?
I'm really confused here, any lead would help, many thanks.