We need a quantity that conveys information about the direction of the radius vector and the direction of the force vector.
Two vectors determine a plane. So we need a quantity that specifies a plane. One way to do that is to specify the vector normal to the plane. That's what the cross product does for us. There is an ambiguity as to direction: there are two normal vectors to a plane. We solve this by choosing one arbitrarily: we decide by convention to use the right-hand rule.
There are other ways to represent a torque that some would argue are more natural, for example, the bivector. These other ways are usually extensible to dimensions higher than three, whereas the cross product works only in three dimensions. Well, we live in a world having three spatial dimensions. That fact, and years and years of usage and tradition, has cemented the cross product into our toolbox.
The cross product has a few oddities associated with it, but it does the job. Some people think we should do away with the cross product. It might be nice to do that and use a more natural mathematical construct, but trying to make a change like that is like rolling a very large boulder up a hill.