TITLE: Which quadratic forms on $\Lambda^2 V$ come from quadratic forms on $V$?
QUESTION [15 upvotes]: Let $V$ be a finite dimensional vector space, say over $\mathbf R$.  Let $g \in S^2 V^*$ be a quadratic form on $V$.  Then $g$ induces a quadratic form $\Lambda^2 g \in S^2 \Lambda^2 V^*$ on $\Lambda^2 V$ defined on simple vectors by $$\Lambda^2 g(u\wedge v, x\wedge y) := \det \begin{bmatrix} g(u,x) & g(u,y) \\\\ g(v,x) & g(v,y) \end{bmatrix}.$$
Is there a nice sufficient and necessary condition for a quadratic form on $\Lambda^2 V$ to be induced from a quadratic form on $V$?

REPLY [4 votes]: Your $\Lambda^2g$ is curvature operator for hypersurface with second fundamental form $g$.
Thus your question can be reformulated the following way

Find a nice algebraic condition for curvature operators which appear as a curvature operators of a hypersurface