Well here is how to prove the Torque-Angular Momentum relation which is the equivalent of F=dp/dt.
We know that
L = r x p
So if we differentiate both sides, we get
dL/dt = dr/dt x p + r x dp/dt
however dr/dt is just v, and as p = mv, therefore v x p = 0. Also using Newton's second law, the second term becomes r x F = T(because that is the definition of torque).
Thus we end up with
dL/dt = T
A few comments. First of all L and T depend not only on the frame of reference you are in but also on the origin you are using . Thus it is very important to specify the origin.
Second, angular momentum shows a neat division just like kinetic energy - we can write
L=r x pcm + Lcm
Where pcm is the momentum of the center of mass and Lcm is the angular momentum about the center of mass in the frame of reference of the center of mass.
Finally, the Torque-Angular momentum relation must be applied in a inertial reference frame. However, there is a very special non-inertial frame in which it also works and that is the center of mass frame(with the origin at the cm). This is a natural consequence of the above splitting of L into two parts.