It is a remarkable fact that the expression \begin{equation} [\vv,\ww](f) = \vv\bigl(\ww(f)\bigr) - \ww\bigl(\vv(f)\bigr) \end{equation} contains no second derivatives, and therefore defines a new vector field $[\vv,\ww]$, which is called the commutator of the vector fields $\vv$ and $\ww$.

Direct computation (which is easiest in a coordinate basis) now establishes the identity \begin{equation} d\alpha(\vv,\ww) = \vv\bigl(\alpha(\ww)\bigr) - \ww\bigl(\alpha(\vv)\bigr) - \alpha([\vv,\ww]) \end{equation} for any 1-form $\alpha$, and applying this identity to an orthonormal basis results in \begin{equation} d\sigma^i(\ee_p,\ee_q) = 0 - 0 - \sigma^i([\ee_p,\ee_q]) \end{equation} so that \begin{equation} g(d\sigma_i,\sigma_j\wedge\sigma_k) = d\sigma_i(\ee_j,\ee_k) = -\sigma_i([\ee_j,\ee_k]) = -\ee_i\cdot[\ee_j,\ee_k] \end{equation}

We can use this expression to recover the standard formula for the (components of the) connection 1-forms in an orthonormal basis. We have \begin{align} 2\Gamma_{ijk} &= g(d\sigma_i,\sigma_j\wedge\sigma_k) - g(d\sigma_k,\sigma_i\wedge\sigma_j) + g(d\sigma_j,\sigma_k\wedge\sigma_i) \nonumber\\ &= -\ee_i\cdot[\ee_j,\ee_k] +\ee_k\cdot[\ee_i,\ee_j] -\ee_j\cdot[\ee_k,\ee_i] \nonumber\\ &= \ee_k\cdot[\ee_i,\ee_j] +\ee_j\cdot[\ee_i,\ee_k] -\ee_i\cdot[\ee_j,\ee_k] \end{align} which is a special case of the Koszul formula for the Levi-Civita connection. Thus, the connection 1-forms can be found by computing commutators of basis vectors.

This version of the Koszul formula simplifies even further in two dimensions, since $i$ and $j$ must be distinct, and $k$ must be either $i$ or $j$. In either case, we obtain \begin{equation} 2\Gamma_{ijk} = 2\ee_k\cdot[\ee_i,\ee_j] \end{equation} or equivalently \begin{equation} \omega_{ij} = \Gamma_{ijk}\,\sigma^k = \ee_k\,\sigma^k\cdot[\ee_i,\ee_j] = [\ee_i,\ee_j]\cdot d\rr \end{equation}