Critical points a Morse function on SO(3)

Question

Let $f_A: SO(3) \to \mathbb{R}$ and is given by $R\mapsto \frac{1}{2}|| A - R ||_F$, where $A \in \mathbb{R}^{3\times 3}$, and the norm is the Frobenius norm. As indicated in [1], the gradient of $f_A$ is $\nabla f_a = A^TR - R^TA$. Suppose A = I. Then the gradient becomes \begin{equation} \label{eq:1} \nabla f_a = R - R^T. \end{equation} I have the following questions:

1. What are the critical points of $f_A$, i.e., for what values of $R\in SO(3)$ the equation $R - R^T = 0$? Clearly, the critical points includes {I,diag(1,-1,-1),diag(-1,-1,1),diag(-1,1,-1)}. Are these the only critical points?
2. Is the set of critical points consist of isolated points. In other words, is $f_A$ a Morse function?

Reference: [1] https://arxiv.org/pdf/2103.14166.pdf