How to plot $SO(3)$ to unit sphere?

Question

I understand that an $SO(3)$ elements corresponds to a $3 \times 3$ rotation matrix, and it could be mapped to $\omega \in \mathfrak{so}(3)$ (lie algebra), quaternion and Euler angles.

But how to plot an $SO(3)$ element to unit sphere (2-sphere or $S_2$)? Since many paper visualised $SO(3)$ on a sphere, e.g.:

1. Figure.1 in Evan, S. Gawlik, Embedding-Based Interpolation on the Special Orthogonal Group

2. Figure 2. in T. Shingel, Interpolation in Special Orthogonal Groups.

Current, I apply a rotation to three axes $\{1,0,0\}$, $\{0,1,0\}$, and $\{0,0,1\}$ respectively and subsequently an $SO(3)$ element corresponds to three points on the sphere instead of one.