Why isn't the derivative of a rotation matrix skew symmetric? Consider the rotation matrix
$$R(\theta) = \begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix}$$
Differentiating $R(\theta)$ with respect to $\theta$ gives a matrix that is not skew symmetric. Aren't infinitesimal rotations heuristically supposed to be skew-symmetric?

The general formula is $$\frac{ dR_{\theta}}{d\theta}=\begin{bmatrix} & -1\\ 1 & \end{bmatrix}R_{\theta},$$ where $R_\theta$ is the rotation matrix you wrote. When $\theta=0$, one has $R_0=I$ and so $\frac{d R_{\theta}}{d\theta}$ is skew-symmetric.
