Let $A$ and $B$ be $n \times n$ matrices. If $B$ is nonsingular, then the polynomial $p(\lambda) = \det (A - \lambda B)$ is clearly the characteristic polynomial of $B^{-1}A$ times $\det B$.
What is known about the degree and the leading order coefficient of this polynomial if $B$ is singular? I am especially interested in the case when $B$ is skew-symmetric and $n$ is odd.