In the proof of the theorem that there is a unique linear isomorphism $\star:\bigwedge^k(E)\to\bigwedge^{n-k}$ on p.4 in Bleecker's Gauge Theory and Variational Principles he says
For $\gamma\in\bigwedge^{n-k}(E)$, define $\varphi_\gamma:\bigwedge^k(E)\to\mathbb{R}$ by $\varphi_\gamma(\alpha)\mu＝\alpha\wedge\gamma$. We can prove that if $\varphi_\gamma(\alpha)＝0$ for all $\alpha$, then $\gamma＝0$. Thus, $\gamma\mapsto\varphi_\gamma$ defines a one-to-one linear map $\bigwedge^{n-k}(E)\to\bigwedge^{k}(E)^\wedge$. Since $\dim\bigwedge^k(E)＝\dim\bigwedge^{n-k}(E)$ this map is an isomorphism.
What is meant by $\bigwedge^{k}(E)^\wedge$?
This is notation for the dual of $\bigwedge^kE$, which is often denoted $\left(\bigwedge^kE\right)^*$.