How could you use a wrench to explain why a x b = - (b x a) for any vectors a and b?

Let P be the point at which the nut is fixed on this plane of the paper which moves perpendicular to the plane of paper on rotation obeying right hand system.

PA be the wrench on the plane of paper .

Let AB be the line on the  plane of paper you apply a force F   on the wrench making an angle PAB =  x  , which is measured  in the sense of PAB (anti clock wise.)

Then the the moment of the force  T about the point P

= PA X F= ( Perpedicular distance from the point P to the line AB)*|F|= PA*|F| sin(anglePAB)  -> the bolt moves in a direction perpendicular to the plane of paper towards us. This similar to aXb =|a|*|b|sin (angle measured from a to b)

Now the bolt is made to rotate in BAP direction. The angle measured in this sense is  oppsite of ABP. Hence angle BAP =-x.

b X a  is similar to  keeping the same magnitude of the force, the same lenth of PA but the sense direction  of movement of the wrench is BAP . So the angle is BAP  as opposed to the ABP. This makes nut move away  in a direction perpendicular to the plane of paper.

Therefore bXa = |b||a| sin (angle measured from b to a)