This question already has an answer here:
From Humphreys' Introduction to Lie Algebras and Representation Theory, written in parentheses:
(It can be shown that even dimensionality is a necessary condition for existence of a non-degenerate bilinear form satisfying $f(v,w)=-f(w,v)$.)
Why is this? Suppose we have odd dimensionality. So $n$ is odd. We have an $n\times n$ matrix $S$ with independent rows such that $v^TSw=-w^TSv$ for all vectors $v,w$. How does this provide a contradiction?