Before, I ask similar this. But here I change question settings since it was incomplete. I hope receive good ideas.
Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as the dual space of $E$ and $E^{**}:=(E^*)^*$ is the double duall of $E$. ($\dim E=n$)
$Alt^P(E):=\{ \alpha\colon \overbrace{E\times\cdots\times E}^{p- times}\rightarrow \mathbb R\ \ , \alpha \text{ is alternating multilinear map}\}$ $Alt^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\ \ , u \text{ is alternating multilinear map}\}$
I wish to prove that there exist an $\textbf{unique}$ bilinear form $B\colon Alt^p(E^*)\times Alt^p(E)\rightarrow\mathbb R$
such that for all $ f_i\in E^* \ , u_j\in E^{**}\stackrel{\scriptsize{isomorph}}\simeq E \qquad i,j=1,\cdots,p$, it has the following rule:
$B(u_1\wedge\cdots\wedge u_p,f_1\wedge\cdots\wedge f_p)=\det[f_i(u_j)]$.
$\wedge$ is wedge product between alternating maps i.e. $\alpha\wedge\beta(v_1,\cdots,v_{p+p})=\dfrac{1}{p! p!}\displaystyle\sum_{\sigma\in S_{p+p}}sign(\sigma)\alpha(v_{\sigma(1)},\cdots,v_{\sigma(p)})\beta(v_{\sigma(p+1)},\cdots,v_{\sigma(p+p)})$
How can I do this?