Let $\theta_X$ be a $1$-form. Petersen's "Riemannian Geometry" says the following on pg 24:
$d\theta_X(\partial_k,\partial_l)=\partial_kg(X,\partial_l)-\partial_lg(X,\partial_k)-g(X,[\partial_k,\partial_l])$
How is this? Can I get a reference for this fact?
If $\theta_X$ is the 1-form defined by $Y\mapsto \theta_X(Y)=g(X,Y)$, then this is a direct application of the formula for the exterior derivative of a 1-form $\omega$: $$d\omega(X,Y)=X\omega(Y)-Y\omega(X)-\omega([X,Y]),$$ and you can find a proof for this one in John M. Lee's Introduction to smooth manifolds, Proposition 14.29.