$$ \begin{aligned} \theta&=\omega t, \end{aligned} $$
$$ \begin{aligned} \vec r&=r\cos\theta\i+r\sin\theta\j\\ &=r\cos(\omega t)\i+r\sin(\omega t)\j\taag1\\ \end{aligned} $$
$$ \begin{aligned} \vec v&=\dyt{\vec r}\\ &=\dt\bra{r\cos(\omega t)\i+r\sin(\omega t)\j}\\ &=-r\omega \sin(\omega t)\i+r\omega \cos(\omega t)\j\taag2\\ \end{aligned} $$
$$ \begin{aligned} \vec \omega &= \omega \k\taag3 \end{aligned} $$
$$ \begin{aligned} \vec \omega \times \vec r&= \omega \k \times \bra{r\cos(\omega t)\i+r\sin(\omega t)\j}\\ &=r\omega \sin(\omega t)\(\k\times\j\)+r\omega \cos(\omega t)\(\k\times\i\)\\ &=r\omega \sin(\omega t)\(-\i\)+r\omega \cos(\omega t)\(\j\)\\ &=-r\omega \sin(\omega t)\i+r\omega \cos(\omega t)\j\\ &=\vec v \end{aligned} $$