Formula for the interior product of a p-form Let $X: \mathbb{R}^n \to \mathbb{R}^n$ be a vector field and let $$\omega = \sum_{i_1 < \dots < i_p} f_{i_1\dots i_p} dx_{i_1} \wedge \dots \wedge dx_{i_p}$$ be a $p$-form over $\mathbb{R}^n$. I am interested in a formula for the interior product $i_X\omega$.
(Choose $n=3$ and $X(x,y,z) = (z,y,-x)$ for all examples below.)
$p = 1$:
\begin{align}
\omega &= \sum_i f_i dx_i &\Rightarrow \quad i_X\omega &= \sum_i f_iX_i\\
\omega &= dx + zdy + zdz &\Rightarrow \quad i_X \omega &= X_x + zX_y + zX_z\\
& & &= z + yz - xz
\end{align}
$p = 2$:
\begin{align}
\omega &= \sum_{i < j} f_{ij} dx_i \wedge dx_j &\Rightarrow \quad i_X\omega &= \sum_{i < j} f_{ij} \left( X_i dx_j - X_j dx_i \right)\\
\omega &= dx \wedge dy + dx \wedge dz &\Rightarrow \quad i_X \omega &= X_xdy - X_ydx + X_xdz - X_zdx\\
& & &= zdy - ydx + zdz + xdx \\
& & &= \left( x - y \right)dx + zdy + zdz
\end{align}
$p = 3$:
\begin{align}
\omega &= \sum_{i < j < k} f_{ijk} dx_i \wedge dx_j \wedge dx_k &&\Rightarrow \quad i_X\omega =\ ???\\
\omega &= 2zdx \wedge dy \wedge dz &&\Rightarrow \quad i_X\omega =\ ???
\end{align}
Questions

*

*Are my formulas and examples for $p=1$ and $p=2$ correct?

*What is the formula for $p \geq 3$?

 A: The formulas and examples for $p=1$ and $p=2$ are correct.
$p = 3$:
\begin{align}
\omega &= \sum_{i < j < k} f_{ijk} dx_i \wedge dx_j \wedge dx_k\\
\Rightarrow \quad i_X\omega &= \sum_{i < j < k} f_{ijk} \left( X_i dx_j \wedge dx_k - X_j dx_i \wedge dx_k + X_k dx_i \wedge dx_j \right)\\\\
\omega &= 2zdx \wedge dy \wedge dz\\
\Rightarrow \quad i_X\omega &= 2z\left( X_x dy \wedge dz - X_y dx \wedge dz + X_z dx \wedge dy \right)\\
&= 2z\left( zdy \wedge dz - ydx \wedge dz - xdx \wedge dy \right)\\
&= -2xzdx\wedge dy - 2yzdx \wedge dz + 2z^2dy\wedge dz
\end{align}
$p \in \mathbb{N}$:
\begin{align}
\omega &= \sum_{i_1 < \dots < i_p} f_{i_1\dots i_p} dx_{i_1} \wedge \dots \wedge dx_{i_p} \\
\Rightarrow \quad i_X\omega &= \sum_{i_1 < \dots < i_p} f_{i_1\dots i_p} \sum_{j=1}^p (-1)^{j-1} X_{i_j} dx_{i_1} \wedge \dots \wedge dx_{i_{j-1}} \wedge dx_{i_{j+1}} \wedge \dots \wedge dx_{i_p}
\end{align}
A: For $p = 3$:
$$\omega = \sum_{i < j < k} f_{ijk} dx_i \wedge dx_j \wedge dx_k \quad \Rightarrow \quad i_X\omega =\ \sum_{i < j < k} f_{ijk} (X_i dx_j \wedge dx_k - X_j dx_i \wedge dx_k + X_k dx_i \wedge dx_j)$$
Generally, let $\omega$ be a $k$-form $\alpha_1 \wedge \alpha_2 \wedge \dots \wedge \alpha_k$, where $\alpha_i$'s are $1$-forms and $X$ be vector field. Then, $$i_X\omega = \sum_{i=1}^{k} (-1)^{i-1} \alpha_i(X)\cdot\alpha_1\wedge\dots\wedge\alpha_{i-1}\wedge\alpha_{i+1}\wedge\dots\alpha_k$$
