Usually this most general case of a generally accelerated reference frame occurs for the treatment of a rigid body. So in the following we think of this moving frame as fixed in the interior of a rigid body. Also all vectors are meant as ##\mathbb{R}^3## column vectors with respect to the Cartesian bases for the inertial frame (vector components ##\vec{x}##) and the body-fixed frame (vector komponents ##\vec{x}'##). Then we have
$$\vec{x}=\vec{R}+\hat{D} \vec{r}'.$$
Here ##\vec{R}## is the vector pointing from the origin of the inertial frame to an arbitrary point fixed within the body which is the origin of the body-fixed reference frame (written in components wrt. the inertial basis), ##\vec{r}'## is the vector (components wrt. the body-fixed basis) from the body-fixed origin to an arbitrary other point in the body. ##\hat{D} \in \mathrm{SO}(3)## is a rotation matrix (usually parametrized with three Euler angles). Then the components of the velocity in the inertial frame are given by
$$\vec{v}=\dot{\vec{x}}=\dot{\vec{R}} +\dot{\hat{D}} \vec{r}'. \qquad (*)$$
Here we have used that ##\vec{r}'## is fixed within the body, i.e., for an observer in the body-fixed reference frame it's a constant in time. Now we want to express this equation in terms of components wrt. the inertial basis. Now we have
$$\dot{\hat{D}} \vec{r}'=\dot{\hat{D}} \hat{D}^{-1} \vec{r}.$$
Since ##\hat{D} \in \mathrm{SO}(3)## we have ##\hat{D}^{-1}=\hat{D}^{\mathrm{T}}## and thus
$$(\dot{\hat{D}} \hat{D}^{-1})^{\mathrm{T}} = (\dot{\hat{D}} \hat{D}^{\mathrm{T}})^{\mathrm{T}} = \hat{D} \dot{\hat{D}}^{\mathrm{T}}.$$
On the other hand
$$\hat{D} \hat{D}^{\mathrm{T}} \hat{D}=1 \; \Rightarrow \; \dot{\hat{D}} \hat{D}^{\mathrm{T}} =-\hat{D} \dot{\hat{D}}^{\mathrm{T}},$$
i.e., ##\dot{\hat{D}} \hat{D}^{-1}## is an antisymmetric matrix and thus we can write
$$(\dot{\hat{D}} \hat{D}^{-1})_{jk} = \epsilon_{jlk} \omega_l.$$
Plugging this into (*) gives
$$v_j=V_{j} + \epsilon_{jlk} \omega_l r_k=V_{j} + (\vec{\omega} \times \vec{r})_j \; \Rightarrow \; \vec{v}=\vec{V}+\vec{\omega} \times \vec{r}.$$
Here ##\vec{V}=\dot{\vec{R}}## is the tranlational velocity of the body against the inertial frame, described by the velocity of the arbitrarily fixed point in the body, which is the origin of the body-fixed reference frame, and which often is convenient to be chosen as the center of mass of the body. The 2nd term describes the velocity of an arbitrary point of the body given by the relative vector ##\vec{r}## from the body-fixed origin to the point in the body because of the rotation of the body around the body-fixed origin relative to the inertial frame, and ##\vec{\omega}## is the momentary angular velocity of this rotation (which can be expressed in terms of Euler angles when needed).
