Why are order-k differential forms sections of the kth exterior power of the cotangent bundle?

The question I ask is in the title. This should be quite well-known, and in fact probably I am going to get the response that it is the definition. To convey my confusion, I have to convey my understanding of what is a differential form and what is the contangent bundle. To simplify things, we assume that our whole setup is immersed in the Euclidean space.

$1$. Differential forms.

We take the definitions from Rudin, Principles of Mathematical Analysis. This is for an open set in $mathbb R^n$.

Suppose $E$ is an an open set in $mathbb R^n$. A $k$-surface in $E$ is a differentiable mapping $Phi$ from a compact subset $D subset mathbb R^k$ into $E$. $D$ is called the parameter domain of $Phi$ consisting of points $mathbf u = ( u_{i_1}, cdots , u_{i_k} )$.

A differential form of order $k geq 1$ in $E$ is a function $omega$, symbolically represented by the sum

$$omega = sum a_{i_1, cdots , i_k}(mathbf x) dx_{i_1}wedge cdots wedge dx_{i_k}$$

where the indices $i_1, cdots , i_k$ range independently from $1$ to $n$, and so that $omega$ assigns to each $k$-surface $Phi$ in $E$ a number$omega(Phi) = int_Phi omega$
, according to the rule

$$int_Phi omega = int_D sum a_{i_1, cdots , i_k}(Phi((mathbf{u})) frac{partial ( x_{i_1}, cdots , x_{i_k})}{partial ( u_{i_1}, cdots , u_{i_k})}dmathbf u $$

where $D$ is the parameter domain of $Phi$, and the functions $a_{i_1}, cdots, a_{i_k}$ are assumed to be real and continuous in $D$.

So in the above definition the differential $k$-form is a certain integral for functions on compact $k$-surfaces. Thus a differential form can be treated as a measure for the $k$-surfaces, which can be integrated.

$2$. Cotangent bundle

We take this from wikipedia.

Let $M times M$ be the Cartesian product of $M$ with itself. The diagonal mapping $Delta$ sends a point $p$ in $M$ to the point $(p,p)$ of $M times M$. The image of $Delta$ is called the diagonal. Let $mathcal{I}$ be the sheaf of germs of smooth functions on $M times M$ which vanish on the diagonal. Then the quotient sheaf $mathcal{I}/mathcal{I}^2$ consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The cotangent sheaf $Omega$ is the pullback of this sheaf to $M$.

Now, Def 2: A differential form $k$-form $omega$ is a section of $wedge^k Omega$.

Question.

We consider an open set in the Euclidean space and look at the two definitions. A priori, to my eyes, both appear to be different things. It needs to be proved that they are the same. Please help me out with a reference with the required proofs.