The definition that I know of is the vorticity $\omega$ is the curl of the velocity $u$. Now I'm reading a note saying $\omega$ is defined to be the $d\times d$ antisymmetric matrix: $$\omega = \frac{1}{2}[\nabla u - (\nabla u)^{T}]$$ especially in $2D$, $\omega = \partial_{1}u_{2}- \partial_{2}u_{1}$.
I couldn't derive why the 2D form is related to the definition by antisymmetric matrix. Thank you.
In dimension two, every antisymmetric matrix is of the form $$ \begin{pmatrix} 0 & a \\ -a & 0\end{pmatrix}. $$ In particular, it is a one-dimensional space. So there is little harm in making the identification $$ a \leftrightarrow a \begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}. $$ Indeed, if you actually computed the difference of the matrices, since $ \nabla u =\begin{pmatrix} \partial_1 u_1 & \partial_2 u_1 \\ \partial_1 u_2 & \partial_2 u_2\end{pmatrix}, $ you would see that
$$ \omega_{matrix} = \frac12 \omega_{scalar}\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix} $$ the factor of $\frac12$ is frequently inconsequential.