Inner product of two kforms

Given two $k$-forms $\alpha$ and $\beta$, return the inner product $\left\langle\alpha,\beta\right\rangle$. Here our underlying vector space $V$ is $\mathcal{R}^n$.

The inner product is a symmetric bilinear form defined in two stages. First, we specify its behaviour on decomposable $k$-forms $\alpha=\alpha_1\wedge\cdots\wedge\alpha_k$ and $\beta=\beta_1\wedge\cdots\wedge\beta_k$ as

$$\left\langle\alpha,\beta\right\rangle=\det\left( \left\langle\alpha_i,\beta_j\right\rangle_{1\leq i,j\leq n}\right)$$

and secondly, we extend to the whole of $\Lambda^k(V)$ through linearity.