Why are rotation matrices represented by the unit sphere?

Question

I have heard two statements:

1. "Although rotation matrices have 9D scalars, they only require 3D elements".
2. Rotation matrices are represented by the unit sphere.

Although I understand point 1), in the sense that the 3 unitary and 3 orthogonality constraints mean only 3 more scalars are needed to solve for the 9 scalars... what does the space of these 3 scalars look like geometrically?

Would it represent the unit sphere? Why or why not?

If not, how else can rotation matrices be thought of as the "unit sphere", especially considering rotation matrices are 9D matrices and not 3D points.

Answer 1

The spin group of Wikipedia rotation paragraph explains well the link between $\text{SO}(3)$ the group of 3D rotations or special orthogonal group and the sphere.