On pg. 377 in Hoffman and Kunze's Linear Algebra(Second Edition) Theorem 6 reads:
Let $V$ be an $n$-dimensional vector space over a subfield of the field of complex numbers, and $f$ be a skew symmetric bilinear form on $V$. Then the rank $r$ of $f$ is even, and if $r=2k$, then there is an ordered basis for $V$ in which the matrix of $f$ is the direct sum of the $(n-r)\times (n-r)$ zero matrix and $k$-copies of the $2\times 2$ matrix $$ \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix} $$
My question is whether or not the theorem is true if the base field of $V$ is just any field, not necessarily a subfield of the field of complex numbers.
I couldn't find any place in the proof where anything special about the base field was assumed.
Thanks to Gregory Grant for suggesting a correction.