  We show that the set of $m \times m$ complex skew-symmetric matrix
polynomials of odd grade $d$, i.e., of degree at most $d$, and (normal) rank at
most $2r$ is the closure of the single set of matrix polynomials with the
certain, explicitly described, complete eigenstructure. This complete
eigenstructure corresponds to the most generic $m \times m$ complex
skew-symmetric matrix polynomials of odd grade $d$ and rank at most $2r$. In
particular, this result includes the case of skew-symmetric matrix pencils
($d=1$).
