Why $\mathbf{R}\mathbf{v}\cdot\mathbf{R}(\mathbf{\omega}\times \mathbf{x}) == \mathbf{v}\cdot(\mathbf{\omega}\times \mathbf{x})$

Question

Suppose $\mathbf{R}\in\mathrm{SO}(3)$, $\mathbf{\omega}\in\mathfrak{so}(3)$ and $\mathbf{R}=\exp(\mathbf{\omega}_{\times})$. We also have two arbitrary 3-D vectors $\mathbf{x},\mathbf{v}\in\mathbb{R}^3$

Can anyone help to explain why $$ \mathbf{R}\mathbf{v}\cdot\mathbf{R}(\mathbf{\omega}\times \mathbf{x}) == \mathbf{v}\cdot(\mathbf{\omega}\times \mathbf{x}) $$

Many thanks!