Cross product uniqueness

Question

I have following relationship between vectors

$A_1'(t)=\psi(t)\times A_1(t) \tag1$ $A_2'(t)=\psi(t)\times A_2(t) \tag2$ $A_3'(t)=\psi(t)\times A_3(t) \tag3$

Given Data

1. " ' " means derivative with respect to t
2. $\times $ means cross product
3. $A_1(t),A_2(t),A_3(t),\psi(t)$ are vectors
4. $A_1(t),A_2(t),A_3(t)$ forms an orthogonal frame.Means they are perpendicular unit vectors

Question

1. Assume we are some how able to generate $A_1'(t) $ ,$A_1(t)$ and figured out an $\psi(t)$ which satisfies equation 1,2, and 3.

   a. Can we say $\psi(t)$ is the angular velocity of the frame with repect to time ?

   b. Can we say $\psi(t)$ is unique?