Finding a specific Rotation matrix given a known vector

rotations

You seem to be looking for the Direction Cosine Matrix (DCM):

Here is a good introduction: http://www.starlino.com/dcm_tutorial.html

Update:

The Rodrigues Formula (and DCM) is giving you the rotation along the shortest path from one frame to the other. You can visualize the path of rotation as an arc on the unit sphere from the one point on the sphere given by vector $P_1$ to other point on sphere given by vector $P_2$. Being both $P_1$ and $P_2$ unit vectors.

Of course there are several paths in the sphere connecting $P_1$ and $P_2$. Thus exist one different matrix per each corresponding path. For more details see the answer to this question:

https://stackoverflow.com/questions/37494882/is-the-rotation-matrix-unique-for-a-given-rotation

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