See the concept of Torque was introduced to deal with the cases in which different parts of the body moves differently or precisely for rotatory motions.

If any force is applied on an object (at a point other than it's center of mass ) then the body will show some rotational motion because the force is not distributed equally. So to find the amount of the effect (i.e. rotational effect) of the force at a point, we need to define its distance from the point of application of the force and also the direction of the angular acceleration is obtained by screw rule and so the cross product sets good here.

Suppose we have a rod hinged at one of its ends and some force is applied near to the other end. The point closer to the hinged end moves in a smaller circle and this means it has more curvature and more angular acceleration and this holds good with the given formula because it's distance from point of application of force is more. 

Also,  the amount of force determines the amount of rotation.

So , the formula for Torque holds good in all scenarios.

**NOTE:** the force due to hinge is ignored.