By defintion, the Hodge star of a $p$-form $\omega_{a_1\cdots a_p}$ on a $n$-dimensional manifold is given by
$*\omega_{b_1\cdots b_{n-p}}=\frac{1}{p!}\omega^{a_1\cdots a_p}\epsilon_{a_1\cdots a_pb_1\cdots b_{n-p}}$
Here, I assume the manifold is equipped with a metric $g_{ab}$ and the volume element is compatible with the metric, ie., $\epsilon_{a_1\cdots a_n}=\sqrt{|g|}\tilde\epsilon_{a_1\cdots a_n}$ with $\tilde\epsilon_{a_1\cdots a_n}=+1$ when $a_1,\cdots,a_n$ is an even permutation of $1,2,\cdots,n$; $=-1$ when old permutation; $=0$ when there are repeated indices.
From this formula, I guess we can calculate the Hodge dual by first take the tensor product of $\omega_{a_1\cdots a_p}$ with $\epsilon_{a_1\cdots a_n}$ and then contract the first $p$ indices. Am I correct?