wedge product of standard represention

Question

For symmetric group $S_{n}$, we have standard representation $V$. We can prove that wedge product $\wedge^{p}V$ is irreducible representation of $S^{n}$ for any $p$. Since we have one-to-one correspondence between irreducible representations of $S_{n}$ and Specht modules $S^{\lambda}$. I was wondering how can we prove that $S^{n-p,1^{p}}\cong \wedge^{p}V$. I can check that dimensions of these two spaces are the same. But I have a hard time to find a proper map between these two spaces.