It is a shame that this question is skipped in almost all modern textbooks! Except Nakahara I know of no other modern textbook.

There's a general **contortion operation**, sometimes called the **Schouten braces** in physicists' index calculus:
$$T_{\{abc\}} := T_{abc} - T_{bca} + T_{cab}$$
See Schouten, p.132 formula (3.7). (This looks a bit unnatural/confusing to me: The last + should be a -.) (From this combined with index juggling and slot mishmash some of the deepest calculatorial confusions in theoretical physics have repeatedly arisen...).

The Schouten braces can also be applied to the nonmetricity tensor $\nabla g$:

**Theorem:** Let $\nabla$ be a (Koszul) covariant derivative with torsion $T$. Let $g$ be an arbitrary pseudo-Riemannian metric with Levi-Civita connection $\nabla^\circ$. There are 2 $g$-dependent contortion operators which embody Schouten braces plus index juggling, $C_1^g$ and $C_2^g$, whose details depend on your favorite variant of total covariant Koszul derivative, such that
$$\nabla=\nabla^\circ+\frac{1}{2}C_1^g\cdot\nabla g+C_2^g\cdot T$$
(Thus $C_2^g\cdot T$ is +/- the contorsion, i.e. the **contor**tion of tor**sion**.)

The only other serious reference I have for this not so hard theorem is *J.A.Schouten, Ricci-Calculus 2nd ed.*, p.132 formula (3.5).