Say you have a cross product a x b = c. Can you intepret this as the vector b spinning about the vector a. If the angle between them is close to zero then vector b is close to zero and is spinning slowly, if the angle between them is close to 90, then vector b is spinning close to maximum. If b is longer than it will spin faster since it's head will have a larger moment.
And the vector c shows the direction vector b is moving at any point as its rotating about a. And the faster b is rotating, the faster it is moving in that direction, therefore the larger is c.
This is the way I interpret the cross product as rotational motion. Is it sufficient or am I glancing over something?
The angular velocity $\;\vec \omega\;$ of a particle at positional displacement $\;\vec r\;$ from the POV moving at (instantaneous) linear velocity $\;\vec v\;$ relative to the POV is indeed calculated as: $$\vec \omega = \frac {\vec r\times \vec v}{\lvert \vec r\rvert^2}$$
(NB: The Point Of View is typical taken as the centre of mass of the system, for convenience.)