Broad Context
As the title suggests, I'm reading Rudin's POMA (3e) and have gotten to chapter 10. Miscellaneous online resources have not been as useful as always, since Rudin (notably, I think) defines differential forms without making any mention whatsoever of manifolds, tensor products, or the exterior algebra (which are all things I've found via. google, and which I do not yet know about). Presumably, Rudin's definition should be understandable without any of these concepts, or else he would have had to introduce them.
Narrow Context
I expect half of the problem will be notation. Here are the definitions I'm working with.
$\mathbf{\mathscr{C}'}$-Mapping: Let $X$ and $Y$ be normed vector spaces over $\mathbb{C}$. Let $B(X,Y)$ denote the set of all bounded linear mappings $X \to Y$. Let $A$ be an open subset of $X$ (with distances measured in terms of the norm of $X$) such that every $a \in A$ is a limit point of $A$. Let $f : A \to Y$ be a map. $f$ is said to be $\mathscr{C}'$ in $A$ if $f$ is Frechet differntiable on $A$ and (letting $\mathcal{D} f(x)$ denote the derivative) the map $A \to B(X,Y) : x \mapsto \mathcal{D} f(x)$ is continuous on $A$ (with distances in $B(X,Y)$ measured in terms of the operator norm).
Full disclose, I've expanded the above definition to a slightly more general context than Rudin presents (definition 9.20 POMA).
$\mathbf{k}$-Surface in $\mathbf{A}$: Let $n$ and $k$ be positive integers. Let $A$ be an open subset of $\mathbb{R}^n$. Rudin defines a $k$-surface in $A$ to be a $\mathscr{C}'$ mapping $K \to A$, where $K$ is a compact subset of $\mathbb{R}^k$.
In particular, Rudin says we are to confine our attention to the situation in which $K$ in the above definition is either a $k$-cell (by which it is meant a Cartesian product of $k$ non-degenerate closed intervals of real numbers), or is the standard $k$-simplex. Note, the above is Def 10.10 and the below is Def 10.11 in POMA.
Differential Form of Order $\mathbf{k \geq 1}$: Let $A \subset \mathbb{R}^n$ be open. Let $\Omega(A)$ denote the set of all $k$-surfaces in $A$ (this is my own notation). Let $\phi_1,\ldots,\phi_n$ denote the component functions of a given $\Phi\in\Omega(A)$. A differential form of order $k$ is a function $\omega : \Omega(A)\to\mathbb{R}$ determined by the rule $$ \Phi \mapsto \int_{\text{dom}(\Phi)} \sum a_{i_1}\cdots\hspace{0.5mm}_{i_k}\big(\Phi(\mathbf{u})\big) \det \big( \mathcal{D} (\phi_{i_1}(\mathbf{u}),\ldots,\phi_{i_k}(\mathbf{u})) \big) \mathrm{d} \mathbf{u} $$ where the integral is a classical (what I would call iteratively defined) multiple integral over the compact subset dom$(\Phi)$ of $\mathbb{R}^k$ (that is, the domain of $\Phi$). Also, the indices $i_1,\ldots,i_k$ "range independently from $1$ to $n$," and "the functions $a_{i_1}\cdots\hspace{0.5mm}_{i_k}$ are assumed to be real and continuous" on $A$. Moreover, Rudin says that the above rule is "symbolically represented by the sum" $$ \omega = \sum a_{i_1}\cdots\hspace{0.5mm}_{i_k}(\mathbf{x})\hspace{1mm} \mathrm{d}_{x_1} \wedge \cdots \wedge \mathrm{d}_{x_k}. $$
Rudin does not provide any sort of definition of "$x \wedge y$."
The Question
Did I happen to get anything wrong in the above definitions?
Is the summation taken over $i\in\{1,\ldots,n\}$, over $(i_1,\ldots,i_k)\in\{1,\ldots,n\}^k$, or over some other indexing set? In other words, how many of these weirdly indexed functions $a_{i_1}\cdots\hspace{0.5mm}_{i_k} : A \to \mathbb{R}$ are necessary in order to completely determine $\omega$?
What the heck is this wedge notation supposed to (as Rudin says) "symbolically represet?" It would almost seem to me that $\omega$ is entirely determined by the set $A$ and the functions $a_{i_1}\cdots\hspace{0.5mm}_{i_k}$, is it not? Do the $\mathrm{d}_{x_1} \wedge \cdots \wedge \mathrm{d}_{x_k}$ specify a subset of $\{1,\ldots,n\}^k$ over which the summation is to be taken, or something like that?
Again, I'm not familiar with manifolds, nor with tensor products, and Rudin has not developed these subjects, at all. Normally, I prefer rigor to intuition, but as a first step, I'll take what I can get. Suggested supplementary readings are also welcome although, realistically, for the foreseeable future I won't have time to read, e.g., a whole book on a subject which is not part of my school work.
Thanks for your time.
1.
I do not see anything you got wrong. I cannot guarantee that everything is exactly correct, but it all looks essentially correct to me.
2.
You can think of the summation as being taken over $\{1,\dots,n\}^k$. However, if two of the indices $i_{\ell}$ and $i_m$ are equal elements of $\{1,\dots,n\}$, then the Jacobian matrix whose determinant you're taking is singular, so that term of the sum is zero. Additionally, terms in the sum which consist of the same subset taken in a different order will either have the same determinant (if the permutation relating the two orders is even) or opposite determinants (if the permutation relating the two orders is odd). So you can combine those $a_{i_1\dots i_k}$ to get a single term.
That is, you can also think of the summation as being taken only over strictly increasing sequences $1 \le i_1 < i_2 < \dots < i_k \le n$, since all the other terms either vanish or can be combined with a term of this form.
3.
The wedge product represents a generic antisymmetric product on the symbols $d_{x_1}, \dots, d_{x_n}$. That is, the wedge product obeys the algebraic rule $d_{x_\ell} \wedge d_{x_m}=-d_{x_m} \wedge d_{x_\ell}$ and no other rules. This corresponds to the facts about the determinant that we noticed in part 2.
You are correct that the differential form is determined only by the set $A$ and the functions $a_{i_1\dots i_k}$. One way to define differential forms on an open set $A$ in $\mathbb{R}^n$ would be to define them as formal sums
$$ \sum_{1\le i_1<i_2<\dots<i_k\le n} a_{i_1i_2\dots i_n} d_{i_1} \wedge d_{i_2} \wedge \dots \wedge d_{i_k} $$
and then the first formula gives the rule for integrating such a form on a given $k$-surface.
I would guess that Rudin gives the definition in the other order because it corresponds more closely to how differential forms are actually used. Intuitively, you should think of a differential $k$-form as "a thing that you can integrate on $k$-dimensional surfaces", which is exactly what the first map in the definition is doing. The fact that such things can be written in wedge product notation is important (especially for computation) but not as conceptually central.