# Understanding mathematical syntax in SO(3)

In this paper, pp. 2, I found the following differential equation system and a statement:

There are two things not clear to me:

1. the hat map symbol seems to make mathematical symbolics shorter and hat the same meanign as a cross product between two vectors (if I understand it correctly). Ok, but the only hat map I can see is on the Matrix: $\hat{\Omega}$. So what cross product should I consider?!?!?
2. is the hat map really a short way to mean a cross product between vectors?!? Since I m not a mathematical student, I m not sure. Usually you use it in engineering field like a "index" for not repeating the same letter in the text;

Thanks a lot.

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".