Proof idea:
$A$ is a skew symmetric matrix.
$A$ is similar to $A^t$ because every matrix is similar to it's transpose. $$A^t = -A $$ $A$ is similar to $-A$. Let $P_{(\lambda)}$ be the characteristic polynomial. $$P_{(A)}=P_{(-A)}=0$$ (I'm not sure about the above step)
And I also know that: $\overline{P_{(A)}}=\overline{P_{(-A)}}=\overline0=0$
That's where I ran out of ideas. am i in the right direction?
