According to wikipedia, the vector equation describing velocity during circular motion is $ \mathbf{v} = \boldsymbol{\Omega} \times \mathbf{r} $, where $\Omega$ is the axis of rotation, with a magnitude representing the rotational velocity.
However, I'm working in 2 dimensions, and I'd prefer not to throw in a 3-dimensional vector just to make the notation look right. How can I write this formula so that its' valid for 2-vectors? I know I can write it like this, where $\omega$ is a scalar:
$\mathbf{v} = \begin{bmatrix}0 & -1 \\ 1 & 0 \end{bmatrix} \times \mathbf{r} \times \omega = \begin{bmatrix}0 & -\omega \\ \omega & 0 \end{bmatrix} \times \mathbf{r} $
But that feels like even more of a kludge. Which piece of notation am I missing?