On pg 73 of "Tensors, Relativity and Cosmology"
The generalized Stokes's theorem in arbitrary $N$-dimensional space is given by: $$\int_c A_mdx^m=\frac{1}{2}\int_S F_{mn}dS^{mn} \tag{1}$$
where $F_{mn}$ is the curl tensor of the vector $A_m$, $F_{mn}=A_{n,m}-A_{m,n}$ (, denotes covariant differentiation here) and $dS^{mn}$ is the contravariant tensor of an infinteseimal element of the surface $S$ $(dS^{mn}=dx^m \wedge dx^n$).
In the three-dimensional metric space the RHS of (1) is equivalent to the ordinary curl A definition
I tried to expand the RHS of (1) to obtain $$\frac{1}{2} \left(\frac{\partial A_n}{\partial x^m}-\frac{\partial A_m}{\partial x^n} \right)dx^m \wedge dx^n$$ since the Christoffel symbols vanish in the three-dimensional Euclidean metric space.
It appears that $$\frac{\partial A_n}{\partial x^m} - \frac{\partial A_m}{\partial x^n}$$ is the definition of curl A but how do I convert $\frac{1}{2}dx^m \wedge dx^n$ into dS to obtain Stokes's theorem in the ordinary vector notation?