Let $G$ be a finite abelian group. It is well-known that every alternating bihomomorphism $\Omega:G\times G \to \mathbb{C}^\times$ arises as the skew $\kappa/\kappa^T$ of a 2-cocycle $\kappa \in Z^2(G)$, where $\kappa/\kappa^T$ depends only on the cohomology class of $\kappa$.

Is there a similar statement for symmetric bihomomorphisms? I'm looking for something of the form "Every symmetric bihomomorphism on $G$ is of the form $\kappa \cdot \kappa^T$ for a 2-cochain $\kappa$ with property X and two 2-cochains with property X give the same bihomomorphism if they differ by Y"