I don't understand the notation for surface integrals used here by E. Cartan in the work On manifolds with an affine connection.
Let me quote the passage and how he expresses the form.
Imagine a Cartesian frame attached to each point $\vec{m}$ in space, with $\vec{m}$ as the origin. Let $\vec{e_1}, \vec{e_2}, \vec{e_3}$ be the three vectors which together with $\vec{m}$ define this frame. The number of parameters characterizing the frame can be as large as 12. Let us denote these parameters by $u_i$. Under infinitesimal changes of these parameters $\vec{m}$ as well as $\vec{e_1}, \vec{e_2}, \vec{e_3}$ undergo infinitesimal vectorial variations which can be expressed as linear combinations of $\vec{e_1}, \vec{e_2}, \vec{e_3}$:
$$\eqalign{ d\vec{m} = w^1\vec{e_1} + w^2\vec{e_2} + w^3\vec{e_3} \cr d\vec{e_1} = w^1_1\vec{e_1} + w^2_1\vec{e_2} + w^3_1\vec{e_3} \cr d\vec{e_2} = w^1_2\vec{e_1} + w^2_2\vec{e_2} + w^3_2\vec{e_3} \cr d\vec{e_3} = w^1_3\vec{e_1} + w^2_3\vec{e_2} + w^3_2\vec{e_3} \cr }$$
The coefficients $w^i$ and $w^i_j$ are linear combinations of the differentials $d\vec{u_i}$. These twelve Pfaffian forms enable one, in effect, to fix the frame at $\vec{m} + d\vec{m}$ in terms of the given frame at $\vec{m}$. These forms $w^i$ and $w^i_j$ are not arbitrary since the integrals:
$$\eqalign{ \int d\vec{m} \cr \int d\vec{e_i} \cr }$$ ($i = 1, 2, 3$) evaluated along any closed loop are obviously zero. Now, if one transforms these into surface integrals, one obtains:
$$\eqalign{ \int d\vec{m} = \int \int (w^1)'\vec{e_1} + (w^2)'\vec{e_2} + (w^3)'\vec{e_3} + d\vec{e_1}w^1 + d\vec{e_2}w^2 + d\vec{e_3}w^3 }$$
It is the last notation of the surface integral that is pretty obscure to me.
*I would be happy to add a link to the original PDF but due to copy rights I'm not sure this is allowed.