Given a $1$-form $\omega$ on $\Bbb R^n$, is there a connection whose torsion is $T(X,Y)=\omega(X)Y-\omega(Y)X$?

Question

Consider $(R^n, g_0 )$, where $g_0$ is the Euclidean metric, and a differential $1$-form $\omega$ on $R^n$. Can this differential form define a connection on $M=R^n$ such that its torsion is $$T(X,Y)=\omega(X)Y-\omega(Y)X ?$$

Thanks.

Answer 1

Hint In a holonomic basis, the components of the torsion tensor are given by $$T^c{}_{ab} = \Gamma^c_{ab} - \Gamma^c_{ba} ,$$ where $\Gamma_{ab}^c$ are the Christoffel symbols of the connection. Can you choose the Christoffel symbols so that the identity is satisfied? (One can even choose them so that the connection is metric.)