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

I'm sure that the answer is known. My guess is: it should be a linear map $\ell:S^2(\Lambda^2)\otimes S^2(\Lambda^2)\to\mathbb{R}^k$ such that $\ell(q\otimes q)=0$ iff $q=\Lambda^2g$ for some $g$. (The map $\ell$ should be "nice" but I do not know what is it.)

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