Large affine spaces of non-singular matrices

Let K be an arbitrary (commutative) field with at least three elements. It was recently proven that an affine subspace of M_n(K) consisting only of non-singular matrices must have a dimension lesser than or equal to n(n-1)/2. Here, we classify, up to equivalence, the subspaces whose dimension equals n(n-1)/2. This is done by classifying, up to similarity, all the n(n-1)/2-dimensional linear subspaces of M_n(K) consisting of matrices with no non-zero invariant vector, reinforcing a classical theorem of Gerstenhaber. Both classifications only involve the quadratic structure of the field K.