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

Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade