Large affine spaces of symplectic forms

Let F be a field and V be a 2n-dimensional vector space over F.

In a previous article, we have proved that if F has more than 2n-2 elements then the greatest possible dimension for an affine space of symplectic forms on V is n(n-1).

Here, under the same cardinality assumption we study the spaces that have the critical dimension n(n-1).

In particular, if the characteristic of F is not 2 the classification of these spaces up to congruence is reduced to: (1) the classification of nonisotropic quadratic forms over F, up to equivalence and multiplication with a nonzero scalar; (2) the classification of nonisotropic Hermitian forms over all quadratic extensions of F, up to equivalence and multiplication by $-1$.

In particular, for quadratically closed fields it is shown that there is exactly one solution up to congruence.