= Interior contraction of the Riemannian volume form
{title2=$\iota_X\omega_g=\sum_i(-1)^{i-1}f_i\theta_i$}
In a positively oriented <Riemannian orthonormal frame> write $X=\sum_i f_iE_i$, with dual <coframe> $\omega_1,\ldots,\omega_n$. Define the omitted-factor <differential forms>
$$
\theta_i=\omega_1\wedge\cdots\wedge\widehat{\omega_i}\wedge\cdots\wedge\omega_n.
$$
The <Riemannian volume form> is $\omega_g=\omega_1\wedge\cdots\wedge\omega_n$, and its <interior product> is
$$
\iota_X\omega_g=\sum_i(-1)^{i-1}f_i\theta_i.
$$
Inserting $E_i$ into the first argument moves it past $i-1$ factors, producing the sign. For $n=2$, this gives $\iota_X(\omega_1\wedge\omega_2)=f_1\omega_2-f_2\omega_1$.
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