I was wondering if there is a simple formula for the 3 by 3 rotation matrix sending the North pole to a specified point on the sphere, whose coordinate is given by the spherical coordinate ($\theta$, $\phi$).
Edit: I realized that the answer is not a single matrix but rather a one-parameter set of matrices
I assume that by the North Pole You mean vector $(0,0,1)^T$. Notice that there are a lot of such rotations, because we can always begin with rotation that fixes the North pole and then apply our rotation. The rotation around $OZ$ axis is given by the matrix $$R_z(\theta)=\left(\begin{array}{ccc} \cos(\theta) & -\sin(\theta) & 0\\ \sin(\theta) & \cos(\theta) & 0\\ 0 & 0 & 1 \end{array} \right) $$ Rotation around $OX$ axis is given by $$R_x(\theta)=\left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & \cos(\theta) & -\sin(\theta)\\ 0 & \sin(\theta) & \cos(\theta) \end{array} \right) $$ Now we can do the following: first apply arbitrary rotation $R_z(\psi)$, then apply $R_x(\phi)$ to move the North pole to $(0, \phi)$, and then apply $R_z(\theta)$ which moves $(0, \phi)$ to $(\theta, \phi)$ as desired. So the complete answer would be $$R_z(\theta)R_x(\phi)R_z(\psi)$$