Let $A$ be a ring and let $F$ be a finitely generated, free $A$-module. Let $\alpha: F \to \textrm{Hom}_A (F, A)$ be a skew-symmetric ring homomorphism, i.e. $\alpha(x)(y)=-\alpha(y)(x)$ for all $x,y \in F$. Now let $M$ be a submodule of $F$ such that $\alpha(M) \subseteq M^{\perp} =\{ f \in \textnormal{Hom}_A (F, A): M \subseteq \ker(f)\}$.

If $A$ is a field, then we can decompose $\alpha=\beta+\beta^*$ where $\beta: F \to \textrm{Hom}_A (F, A)$ satisfies $M \subseteq \ker(\beta)$ and $\beta^*$ is the adjoint homomorphism. I am interested in the case where $A$ is the polynomial ring in several variables over some field. Are there some conditions on $\alpha$ and $M$ known, such that we can decompose $\alpha$ in such a way?