Suppose $c: I \rightarrow O(n)$ is a smooth curve with $c(0) = \text{Id}_n$, where $\text{Id}_n$ is the nxn identity matrix, and $I \subset \mathbb{R}$, and $O(n)$ is the set of orthogonal $n$x$n$ matrices.
Prove that $c'(0)$ is skew-adjoint, i.e. $c'(0) + c'(0)^T = \mathbf{0}$.
All I can think of is some relations from physics between dot products of vector functions and their derivatives.