Question: A matrix $A$ is called orthogonal iff $A^TA=I$. It is further called rotation matrix if $A^TA=I, \det A=1$. These are well known. But how to bridge the algebraic definite of rotation matrices and their geometric role?

I think the most essential geometric role of a rotation matrix is to rotate every vector in $R^n$ the same rotation angle around the same rotation axis. Therefore, geometrically, a rotation matrix is determined by a rotation angle and and rotation axis. Now the question is actually how to understand the relationship of the following algebraic and geometric definition:

Algebraic: $A^TA=I, \det A=1$
Geometric: rotation angle $\theta$ and rotation axis $v$

In other words, prove

1. Given $A^TA=I, \det A=1$, find the rotation angle and axis.
2. Given rotation angle $\theta$ and rotation axis $v$, prove the corresponding matrix satisfy: $A^TA=I, \det A=1$