An identity of degree 2n − 3 for the n × n skews, n even, and corollaries for standard identities

Let n be a positive integer and let Kn(F,t) denote the space of skew-symmetric matrices (with respect to the transpose involution) matrices over a field F. A theorem of Kostant (n even) and Rowen (n odd) states that Kn(F,t) satisfies the (2n − 2)-fold standard identity s2n-2. In this paper we prove the existence of an identity of degree 2n − 3 for even, and then use this identity to obtain both a new proof and a refinement of the Kostant–Rowen results.