Affine subspaces of antisymmetric matrices with constant rank

Abstract: For every $n \in \mathbb{N}$ and every field $K$, let $A(n,K)$ be the vector space of the antisymmetric $(n \times n)$-matrices over $K$. We say that an affine subspace $S$ of $A(n,K)$ has constant rank $r$ if every matrix of $S$ has rank $r$. Define $${\cal A}{antisym}^K(n;r)= { S \;| \; S \; \mbox{\rm affine subspace of $A(n,K)$ of constant rank } r}$$ $$a{antisym}^K(n;r) = \max {\dim S \mid S \in {\cal A}{antisym}^K(n;r) }.$$ In this paper we prove the following formulas: for $n \geq 2r +2 $ $$a{antisym}^{\mathbb{R}}( n; 2r) = (n-r-1) r ;$$ for $n=2r$ $$a_{antisym}^{\mathbb{R}}( n; 2r) =r(r-1) ;$$ for $n=2r+1$ $$a_{antisym}^{\mathbb{R}}( n; 2r) = r(r+1) .$$