linear algebra – Can I estimate dot{omega} in this equation, even though I cannot directly solve the equation to it?

I am trying to measure/estimate the angular acceleration of an object
$dot{omega} $ from a measurement of it’s acceleration (using an accelerometer) $^{i} {boldsymbol{a}}_m$. As far as I understand, the accelerometer will measure a linear acceleration due to rotation of the object.

From rigid body kinematics, the following relation is know

begin{align*} {^{i} {boldsymbol{a}}_m} & = {^{i} {boldsymbol{a}}_l} + ^{i} dot{{boldsymbol{omega }}}_{i} times {^{i} {{boldsymbol{X}}}_{S_m}} + {^{i} {{boldsymbol{omega }}}_{i}} times left({^{i} {{boldsymbol{omega }}}_{i}} times {^{i} {{boldsymbol{X}}}_{S_m}} right) ; end{align*}

Assuming I know everything in this equation except for $dot{omega}$, I would like to estimate $dot{omega}$.

Unfortunately, the equation cannot directly be solved to $dot{omega}$ since $dot{omega}$ is in a cross product with a vector.

Are there any mathematical tools that can help me estimate $dot{omega}$ given the relation I described above?