Let $V$ and $W$ be two vector spaces. Use $L(V\rightarrow W)$ to represent the vector space of linear map from $V$ to $W$.
It is proved that $\Lambda^p(V^*)\cong (\Lambda^pV)^*$, where $\Lambda$ is the exteror power, which is the case $W=\mathbb{R}$.
The general case seems to be false. But I failed to provide an counterexample. Can anyone help?
Just compute the dimensions. Suppose $\dim V = n$ and $\dim W = m$. On the one hand, $\Lambda^p L (V \to W)$ has dimension $\frac{(n m)!}{p! (n m - p)!}$; on the other hand, $L (\Lambda^p V \to W)$ has dimension $\frac{n! m}{p! (n - p)!}$. Of course, these two quantities are equal when $m = 1$, but already for $m = 2$ we have $\frac{(2 n)!}{p! (2 n - p)!}$ vs $\frac{2 \cdot n!}{p! (n - p)!}$, and the former is in general bigger than the latter. (For instance, take $p = n = 2$.)