In Feynman's lecture on rotations in space, http://www.feynmanlectures.caltech.edu/I_20.html, he introduced the cross product by building upon a definition of torque he had derived in a previous lecture,
$$\tau=xF_y-yF_x$$
He explained that radial distance, $\mathbf{r} = \sqrt{x^2+y^2}$, and force, $\mathbf{F}=\sqrt{F_x^2+F_y^2}$, are just two vectors and that any two vectors can be combined similarly to get a third resultant vector. Rather than try to represent the result on the same plane as the original vectors it makes organizational sense to represent the result in the third dimension, perpendicular to the original vectors. This is the cross product $\mathbf{c = a \times b}$:
$$c_x = a_y b_z - a_z b_y$$ $$c_y = a_z b_x - a_x b_z$$ $$c_z = a_x b_y - a_y b_x$$
Glorious illumination! I had been trying for years to figure out why arbitrarily smooshing vector components together resulted in a vector sticking out of the blackboard somehow equivalent to simple torque. I now understand that the cross product is just a pseudo vector to represent two things that interact orthogonally.
But I still don't feel 100% about about Feynman's original definition for torque. I followed the geometrical proof but I'm hoping there is a straightforward, intuitive, way to understand why $$xF_y-yF_x = \mathbf{rF_{tangential}}$$ If anybody could help me fill in this last hole it would be greatly appreciated.