A necessary condition are the Pluecker relations. Namely, let $W= V^*\otimes V^*\ni g$, then $h=\Lambda^k g$ is decomposable. In detail, $h\in \Lambda^k W$ is decomposable, i.e., $h=g_1\wedge g_2\wedge\dots\wedge g_k$, if and only $i_{\Phi}h\wedge h = 0$ for all $\Phi\in\Lambda^{k-1}W^* = \Lambda^{k-1}(V\otimes V)$. It then remains to ensure that all $g_1$ are the same and are positive definite. See here for various equivalent versions of the Pluecker relations.

A necessary condition are the Pluecker relations. Namely, let $W= V^*\otimes V^*\ni g$, then $h=\Lambda^k g$ is decomposable. In detail, $h\in \Lambda^k W$ is decomposable, i.e., $h=g_1\wedge g_2\wedge\dots\wedge g_k$, if and only $i_{\Phi}h\wedge h = 0$ for all $\Phi\in\Lambda^{k-1}W^* = \Lambda^{k-1}(V\otimes V)$. It then remains to ensure that all $g_1$ are the same and are positive definite. See here for various equivalent versions of the Pluecker relations.