How to calculate angular velocity vector of a moving reference frame represented in another reference frame's axes?

Question

Note: This problem is part of a MATLAB program if that is of any help

2 reference frames {A} and {B}, where {A} is coincident with the origin (0,0,0) AKA fixed frame. B has a 4x4 transformation matrix representing its position and orientation relative to the fixed frame {A} given by T_B.

T_B example at position (1,4,3) rotated 90 in {A} Z axis.

T_b = [ 0 -1 0 1
        1  0 0 4
        0  0 1 3
        0  0 0 1 ]

T_B is transformed over time while {A} is fixed. I can find the instantaneous velocity of {B} at any point in time by simply differentiating each term in the position vector. V in the image

However how would I go about finding the instantaneous angular velocity of {B} ,ω_B, represented in {A} axes the same way the linear velocity is.

For example if {B} follows a random transformation over time as shown in the image (orange line). How would you calculate the angular velocity vector of {B} as seen from the {A} axes?

Visualized problem