For your second question:
For any vector $x$ in 3-space, there is a map on $\mathbb R^3$ that sends $y \mapsto x \times y$. That map is linear, and it's sometimes useful to be able to talk about the _map_ rather than the value of the linear map on some particular vector $y$. So the author has given the map a name -- $\hat{x}$. To make it concrete, if $x = [a, b, c]$, then $\hat{x}$ is multiplication by the matrix: $$ \begin{bmatrix} 0 & -c & b\\\ c & 0 & -a\\\ -b & a & 0 \end{bmatrix}. $$
For the first: I think that the author is treating the angular velocity $\Omega$ as a 3-component _vector_ $(a, b, c)$, and the associated map $\hat{\Omega}$ is therefore "multiplication by the matrix I've written above".