Why is torque the cross product of the radius and force vectors?

I understand the torque vector to be the cross product of the radius (moment arm) and force vectors, but that means the torque would be perpendicular to the radius and force vectors, which makes no sense to me, e.g. a force applied tangent to the surface of a car tire creates a torque along the line of the axle.

I'm pretty sure I am just misunderstanding a simple formula, so I wanted to make sure.

And, when you use the formula for torque, is torque defined as a vector or just a scalar? I would think it would be a vector.