4 dimensional curl as antisymmetric matrix

[purplegreen]

I'm a bit confused. I'm trying to calculate the curl of a 4 dimensional matrix. It's an attempt to use stokes theorem for 4 dimensions.

The curl can be written as a antisymmetric matrix from what I understand with entries,

Mi,j = d Ai/d j - dAj/di

where i and j would be the different coordinates like x, y, z etc... However, from what I understood if you looked the integral about an infinitesimal square in the x-y plane you could work out the integral as:
(dAy/d x - dAx/dy)ΔxΔy

I was informed that this would gives M1,2 ΔxΔy
which would be wrong, you would get -M12 surely?

So how does the curl look in matrix/ tensor form for 4 dimensions?
Hopefully this makes some sense, sorry if it's a slightly confused question.

 

[fresh_42]

In 4 dimensions the curl of a vector field is, geometrically, at each point an element of the $6-$dimensional Lie algebra $\mathfrak{so}(4)$, all skew-symmetric real matrices: $v^*\otimes w - w^*\otimes v$.

You can search for this Lie algebra or Lie group for further information.