For a fixed ordered subset $I=\{i_1,...,i_n\}$ of $\{0,1,...,n\}$, where $i_1<...<i_n$, let $\hat i$ be the unique element of $\{0,1,...,n\} \setminus \{i_1,...,i_n\}$. For such an $I$, let $\sigma_I\in S_{n+1}$ be the permutation that sends $0$ to $\hat i$ and $k \to i_k ,\forall k=1,...,n$ and define $dX^I:=(-1)^{sgn (\sigma_I) } dx_{i_1} \wedge ... \wedge dx_{i_n}$.
Consider an $n$-form $\phi=\sum_{I} f_{\hat i} dX^I$, where $f=(f_0,...,f_n):\mathbb P^n \to \mathbb P^n$ is a morphism, so that each $f_j$, for $j=0,1,...,n$, is a non-zero homogeneous polynomial , and all $f_j$ are of same degree $d\ge 0$.
My question is: How do we compute the Hodge star and Hodge dual of $\phi$ ?