Let $V$ be a real vector space of dimension $n$ with an inner product $g$. Let $e_1,\ldots,e_n\in V$ be an orthonormal basis with respect to $g$. On the $k$th exterior power $\bigwedge^k V$ the induced inner product is characterized by the property that the wedges $e_{i_1}\wedge\cdots\wedge e_{i_k}$ give an orthonormal basis for the induced inner product, where $i_1 < i_2 < \cdots < i_k$.

In particular, the top wedge power $\bigwedge^n V\cong\mathbb{R}$ is generated by the norm 1 vector $\omega := e_1\wedge\cdots\wedge e_n$.

Let $\beta\in\bigwedge^k V$ be a $k$-form. The Hodge dual (or Hodge star) of $\beta$ is by definition the $(n-k)$-form $*\beta$ satisfying: $$\alpha\wedge(*\beta) = g(\alpha,\beta)\cdot\omega$$ for all $k$-forms $\alpha\in \bigwedge^k V$.

I'd like to know if there is a "formula" for $*(\alpha\wedge\beta)$ in terms of $\alpha,\beta,*\alpha,*\beta$ where $\alpha,\beta$ are forms of degrees $k,\ell$ respectively, with $k+\ell < n$.


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