  Given a bilinear form on $\mathbb C^n$, represented by a matrix $A\in\mathbb
C^{n\times n}$, the problem of finding the largest dimension of a subspace of
$\mathbb C^n$ such that the restriction of $A$ to this subspace is a
non-degenerate skew-symmetric bilinear form is equivalent to finding the size
of the largest invertible skew-symmetric matrix $B$ such that the equation
$X^\top AX=B$ is consistent (here $X^\top$ denotes the transpose of the matrix
$X$). In this paper, we provide a characterization, by means of a necessary and
sufficient condition, for the matrix equation $X^\top AX=B$ to be consistent
when $B$ is a skew-symmetric matrix. This condition is valid for most matrices
$A\in\mathbb C^{n\times n}$. To be precise, the condition depends on the
canonical form for congruence (CFC) of the matrix $A$, which is a direct sum of
blocks of three types. The condition is valid for all matrices $A$ except those
whose CFC contains blocks, of one of the types, with size smaller than $3$.
However, we show that the condition is necessary for all matrices $A$.
