Generators of Lie Groups and Angular Velocity

[observer1]

I envision the three fundamental rotation matrices: R (where I use R for Ryz, Rzx, Rxy)

I note that if I take (dR/dt * R-transpose) I get a skew-symmetric angular velocity matrix.
(I understand how I obtain this equation... that is not the issue.)

Now I am making the leap to learning about Lie Algebras and Lie Groups

And I understand that any Rotation matrix can be represented with the exponential map
And with the exponential map, the generator of the map happens to have a form that is skew symmetric and (aside from the coefficients) of the same form as the angular velocity matrix.

Of course. A rotatoin matrix is a change. The skew symmetric angular velocity matrix is a RATE of change.
How is it possible for these to related to each other through an exponential map that does not involve TIME.

 

[LightHero]

The answer is: the exponential map represents an infinitesimal change in R. The angular velocity matrix is a measure of the rate of change of R over a finite amount of time. By taking the derivative of the exponential map, we can obtain an expression for the angular velocity matrix. This is how the two are related.