How do we denote a surface embedded in four dimensions? In three dimensions we can denote a surface in the $xy$-plane with area $1$ as $\vec{S} = 1\hat{z}$. This has many useful applications since we can now take a dot product between a vector field and a surface to find the flux through said surface.
However, when I attempt to think of a similar method for denoting a surface in four dimensions I run into a problem: there are ${4 \choose 2}= 6$  possible planes$^1$, however, there are only $4$ dimensions I can use to represent these planes.
How do mathematicians deal with this problem when extending calculus into higher dimensions?

$^1$ (the $xy$-plane, the $xz$-plane, the $xw$-plane, the $yz$-plane, the $yw$-plane, and the $zw$-plane)
 A: You want to form a wedge product of the differentials (of a parameterized surface) to construct an area element.  The wedge products encode an oriented area element, and do not require any notion of a normal that could be ambiguous in a higher dimensional space.  Such an area element can be described using differential forms, or in geometric algebra.  Here is a rough outline of such a surface encoding in the geometric algebra formalism:
Suppose your surface is parameterized by
$$   \mathbf{x} = \mathbf{x}(u,v).$$
The partials are tangent to the surface
$$\begin{aligned}\mathbf{x}_u &= \frac{\partial {\mathbf{x}}}{\partial {u}}  \\ \mathbf{x}_v &= \frac{\partial {\mathbf{x}}}{\partial {v}}\end{aligned}$$
and you can form a surface area element by wedging the differentials $ d\mathbf{x}_u = \mathbf{x}_u du, \, d\mathbf{x}_v = \mathbf{x}_v dv $ along each of those tangent space directions
$$   d^2 \mathbf{x} = (\mathbf{x}_u \wedge \mathbf{x}_v) \, du dv.$$
If you happen to be in 3D, this area element can be related to the cross product by multiplying by the (trivector) pseudoscalar $ I = \mathbf{e}_1 \mathbf{e}_2 
\mathbf{e}_3 $,
$$   -I d^2 \mathbf{x} = (\mathbf{x}_u \times \mathbf{x}_v) \, du dv.$$
In differential forms, the notation and nomenclature is a bit different (2-form vs. bivector, hodge dual vs. pseudoscalar product) but a lot of the ideas should be similar.
