I'm trying to prove the invariant formula by induction $$d\omega(X_0,\cdots,X_p)=\sum (-1)^i X_i(\omega(\cdots,\hat{X}_i,\cdots))+\sum_{i<j} \omega([X_i,X_j],\cdots,\hat{X}_i,\cdots,\hat{X}_j,\cdots)$$ where $\omega$ is $p$-form and $\hat{X}_i$ means omission. The calculation for $p=1$ is easily done, but I'm stuck with the induction step because I'm very confused by the multiple indices notation when contracting differential forms and vector fields.
I guess one would like to proceed as: suppose the formula holds for $p-1$ form, take $\omega=\alpha\wedge \beta$ where $\alpha$ is $1$-form and $\beta$ is $p-1$ form, one needs $$d(\alpha\wedge\beta)(X_0,\cdots,X_p)=\sum (-1)^i X_i((\alpha \wedge \beta)(\cdots,\hat{X}_i,\cdots))+\sum_{i<j} (\alpha\wedge \beta)([X_i,X_j],\cdots,\hat{X}_i,\cdots,\hat{X}_j,\cdots)$$ the left hand side is $$(d\alpha \wedge \beta -\alpha \wedge d\beta)(X_0,\cdots,X_p)$$ where one should plug in the formula for $1$-form and $p-1$ form. But I am very confused by the contracting process involving permutation of indices.
I know that this theorem could be proved without doing the tedious calculation but if anyone can show it for me, it would be very helpful for me to understand the contracting process. Thanks in advance!