Connection with torsion

Question

Consider a metric connection $\Gamma^{\mu}_{~~\nu\lambda}$ with torsion $$\Gamma^{\mu}_{~~\nu\lambda} = \tilde{\Gamma}^{\mu}_{~~\nu\lambda}+ K^{\mu}_{~~~\nu\lambda}$$ where $\tilde{\Gamma}^{\mu}_{~~\nu\lambda}$ is the Levi-Civita connection and $K^{\mu}_{~~~\nu\lambda}$ is the contorsion tensor. Now, consider the Euler class $e$ computed for each connection. The Chern-Weil homomorphism states that $e(F)-e(\tilde{F})$ is exact where $F$ and $\tilde{F}$ represent the curvature computed for the respective connection. Is there a nice formula in terms of the contorsion tensor for $e(F)-e(\tilde{F})$ ? If not, how such a formula maybe derived ?