Antisymmetric gradient matrix?

[scoobmx]

Does this operator (in 3D):

$$ε_{ijk}∇_k = \begin{pmatrix} 0 & \frac{\partial}{\partial z} & -\frac{\partial}{\partial y}\\ -\frac{\partial}{\partial z} & 0 & \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y} & -\frac{\partial}{\partial x} & 0 \end{pmatrix}$$

have a formal name and a more compact symbolic representation?

 

[Matterwave]

That gives you the curl in 3-D.

It gives you an analogue of the curl in other dimensions. I don't know how much more compact than $\epsilon_{ijk}\nabla_k$ you wanted, but I am not aware of any more compact forms.