Connection matrix in an orthonormal frame is skew-symmetric
= Connection matrix in an orthonormal frame is skew-symmetric
{title2=$A^T=-A$}
In a local <orthonormal frame> $(e_i)$, write $\nabla e_j=A^i{}_j e_i$. <Metric compatibility> and $d\langle e_i,e_j\rangle=0$ give
$$
A^i{}_j+A^j{}_i=0.
$$
Thus the connection one-form of a real metric connection takes values in the <skew-symmetric matrices>.