Conversion of a matrix differential equation for rotation to quaternions

Question

Specifications and Data

1. We have a 3D rotation function $R(t)_{3\times 3}$ and function ${K(t)}_{3\times 3}$, a matrix-valued function that gives skew symmetric matrices as outputs. It has the following form:

$K(t)= \begin{bmatrix}\,0&\!-a_3(t)&\,\,a_2(t)\\ \,\,a_3(t)&0&\!-a_1(t)\\-a_2(t)&\,\,a_1(t)&\,0\end{bmatrix}. \tag 1$ $a_i(s)$ are functions with numerical values, not matrix values unlike the previous case.

2. Let $q(t)$ denote the quaternion of $R(t)$.
3. We have an expression $\frac{\mathrm{d} R(t)}{\mathrm{d}t} = -K(t) R(t).\tag 2$

Question

1. How can we represent the equation(2) in quaternions instead of using 3D rotation matrices? How do we derive that? or else how do we write the equation (2) using $q(t)$ and $\frac{\mathrm{d} q(t)}{\mathrm{d}t}$.

NB : I am not implying doubt about how to solve equation(2). I am just seeking to convert it to a quaternion form.

Thanks for taking time to read my question. Any partial answers are also welcome.