I prefer to use this notation.
The components of a arbitrary vector $\vec{x'}$ in rotating system are transformed to inertial system by this equation:
$$\vec{x}=R\,\vec{x'}\tag 1$$
where R is the transformation matrix between the rotating system and inertial system
The time derivative of equation (1) is:
$$\vec{\dot{x}}=R\,\vec{\dot{x}'}+\dot{R}\,\vec{x'}\tag 2$$
with
$\dot{R}=R\,\tilde{\omega}\quad$
and $\tilde{\omega}=\left[ \begin {array}{ccc} 0&-\omega_{{z}}&\omega_{{y}}
\\ \omega_{{z}}&0&-\omega_{{x}}\\
-\omega_{{y}}&\omega_{{x}}&0\end {array} \right]
$
thus:
$$\vec{\dot{x}}=R\,\vec{\dot{x}'}+R\,\tilde{\omega}\,\vec{x'}\tag 3$$
multiply equation (3) from the left with $R^T$
$$R^T\,\vec{\dot{x}}=\vec{\dot{x}'}+\vec{\omega}\times \vec{x'}$$
thus
$$\boxed{\left(\vec{\dot{x}}\right)_R=\left(\vec{\dot{x}'}\right)_R+
\left(\vec{\omega}\times \vec{x'}\right)_R}$$
where index R means the components are given in the rotating system.