On every Riemannian manifold $M$, we can consider the Hodge $*$-operator, which is characterised by the following formula: $$a \wedge *b = (a,b)\nu.$$ Here $a$ and $b$ are smooth forms on $M$, $(\ ,\ )$ is a metric on $\wedge T^*\!M$ and $\nu$ is the volume form with respect to the Riemannian metric.
My question: Is a formula of $*(a \wedge b)$ known?
I suspect that we can have a formula like "$*(a \wedge b)=(*a)\wedge(*b)$" or "$*(a \wedge b) = *a \wedge b \pm a \wedge *b$". Of course these formulae never hold. (Look at the degree.)