According to Wikipedia,
The Hodge star operator on a vector space $V$ with an inner product is a linear operator on the exterior algebra of $V$, mapping $k$-vectors to $(n-k)$-vectors where $n=\dim V$, for $0 ≤ k ≤ n$. It has the following property, which defines it completely: given two $k$-vectors $α$, $β$, $$\alpha \wedge (\star \beta) = \langle\alpha,\beta \rangle\,\omega$$ where $\langle \cdot,\cdot \rangle$ denotes the inner product on $k$-vectors and $\omega$ is the preferred unit $n$-vector.
But suppose instead that $\alpha$ is a $k$-vector and $\beta$ is a $p$-vector, with $k\ne p$. Then is there a well-known formula for $$\alpha \wedge (\star \beta)$$ like there is when $k=p$? I think $\langle\alpha,\beta \rangle$ is not defined in this case, correct?