Is there a linear injection $ \Lambda^k V^* \otimes \Lambda^k V^* \to \Lambda^k (V^* \otimes V^*)$ which preserves decomposability?

Question

Let $V$ be an $n$-dimensional real vector space, and let $2 \le k \le n-2$.

Definitions

1. We say an element $\omega \in \Lambda^k V$ is decomposable if $\omega=\alpha_1 \wedge \dots \wedge \alpha_k$, for some $\alpha_i \in V$. We say $\omega$ is a power if $\omega=\alpha \wedge \dots \wedge \alpha$ for some $\alpha \in V$.

2. We say an element $h \in \Lambda^k V^* \otimes \Lambda^k V^* \cong \operatorname{Hom}(\Lambda^k V,\Lambda^k V^*)$ is a power if $h=\Lambda^k g$ for some linear map $g:V \to V^*$. Here, $\Lambda^k g:\Lambda^k V \to \Lambda^k V^*$ is the induced map on exterior powers, that is $$ \Lambda^k g(v_1 \wedge \dots \wedge v_k)=g(v_1) \wedge \dots \wedge g(v_k).$$

Question: Is there a linear injection $$ \Lambda^k V^* \otimes \Lambda^k V^* \to \Lambda^k (V^* \otimes V^*)$$ which maps power elements to decomposable elements?

Even better, is there an injection which maps power elements to power elements?

Note: For dimensional reasons, there is always some linear embedding of $\Lambda^k V^* \otimes \Lambda^k V^*$ in $ \Lambda^k (V^* \otimes V^*)$.

Motivation:

This question arose in the context of applying the Plucker relations, to characterise which metrics on exterior powers are induced by metrics at the base. See here.