TITLE: A question on the unit tangent bundle of the sphere and $SO(3)$
QUESTION [6 upvotes]: Let the unit tangent bundle be defined as follows:
$$T^1S^2=\{(p,v)\in \mathbb R^3 \times \mathbb R^3 | |p|=|v|=1 \text{ and } p \bot v \}$$
Let $SO(3)$ be the group of rotations of $\mathbb R^3$. Apparently, $SO(3)$ is in bijection with $T^1S^2$.
My question is:

If $N$ is a point on $S^2$, say the north pole, does the
  rotation in $SO(3)$ moving $N$ to $p$ along $v$ correspond to $(p,v)$
  in $T^1 S^2$?

Put the other way around: 

Does the matrix $(p,v, p \times v)$ corresponding to $(p,v)$ represent
  the rotation around the axis $p$? And if so, is the angle somehow
  represented by $v$?

Later added
The reason why I think there should be geometric meaning to this bijection or at least some insight to be gained is that finding the bijection was an exercise in a book I am reading. 
If there was no insight to be gained the exercise would be more or less purely computational and not very insightful.

REPLY [2 votes]: You can use the poincare representation of $SO(3)$, which gives a  bijective and differentiable mapping between $T_1 S^2$ and $SO(3)$. The  representation is like this: first an element of $SO(3)$ can be identified as an orthonormal frame of $R^3$, and then each point $(p,v)\in T_1S^2$ gives an orthonormal frame: $\{p, v, p\times v\}$.