In many mathematics textbooks the $m$-form on $T_p \mathbb{R}^n$ is written in a compactified notation that I haven't seen before:
$$\omega = \sum_{(1 \le i_1 < \cdots < i_m \le n)}a_{i_1 \cdots i_m} dx_{i_1} \wedge dx_{i_2} \wedge \cdots \wedge dx_{i_m}$$
Which is sometimes compactified even further to
$$ \omega = \sum_{I}a_{I} dx_I \quad ,I = (i_1 , \cdots , i_m)$$
Unfortunately there is always (in every book and online source that I've seen anyways) a dearth of concrete examples for this notation. I want to check and make sure that I'm understanding the notation properly. Are these two sub-examples correct?
(a)
Let's look at $2$-forms in $T_p \mathbb{R}^3$:
$$\omega = a_{i_1 i_2 } dx_{i_1} \wedge dx_{i_2} + a_{i_1 i_3} dx_{i_1} \wedge dx_{i_3} + a_{i_2 i_3} dx_{i_2} \wedge dx_{i_3}$$
For readability, set $i_1 = x$, $i_2 = y$ and $i_3 = z$. Then we set our $dx_\star$ values to our familiar $dx, dy, dz$ notation, and can give our form as
$$\omega = a_{x y } dx \wedge dy + a_{xz} dx \wedge dz + a_{yz} dy \wedge dz$$
$dx, dy$ and $dz$ are the usual covectors that we are familiar with. Some $a_{ij}$ is a scalar, and the $_{ij}$ is some indexing value, similar to how we index $2$-dimensional tensors when doing tensor algebra. That is, $a_{ij}$ can vary if we change $i$ or $j$. Let's keep $dx, dy, dz$ fixed and set $a_{xy} = 5$, $a_{xz} = 7$, and $a_{yz} = 9$. Then
$$\omega = 5 dx \wedge dy + 7 dx \wedge dz + 9 dy \wedge dz$$
Or we can tweak our $a_{ij}$ values and have $a_{xy}' = 2$, $a_{xz}' = 3$, and $a_{yz}' = -13$, which gives
$$\omega' = 2 dx \wedge dy + 3 dx \wedge dz - 13 dy \wedge dz$$
In other words, the way that we generate forms is similar to how we generate vectors in linear algebra, using a basis which spans our vector space and writing any vector $v$ which is in the space as
$$ v = \sum_i a_i e_i$$
(b)
Differential $m$-forms behave similarly, only now instead of being limited to constants we are limited to some $C^\infty$ function that we'll denote and index as $f_I$. Our notation becomes
$$ \omega = \sum_{I}f_{I} dx_I \quad ,I = {(i_1 , \cdots , i_m)}$$
Let's keep the example similar and look at differential $2$-forms in $T_p \mathbb{R}^3$:
$$\omega = f_{x y } dx \wedge dy + f_{xz} dx \wedge dz + f_{yz} dy \wedge dz$$
Choose $f_{xy} = (x^2 + 3x + 2)$, $f_{xz} = -\sin{x}$, and $f_{yz} = \pi^x$. Then
$$\omega = (x^2 + 3x + 2) dx \wedge dy -(\sin{x}) dx \wedge dz + (\pi^x) dy \wedge dz$$
Similarly, choose $f_{xy}' = x$, $f_{xz}' = \cos(x)$, and $f_{yz}' = 4$, then
$$\omega = (x) dx \wedge dy + (\cos x) dx \wedge dz + 4 dy \wedge dz$$
This is again similar to defining vectors in linear algebra, only replace the constants with any $C^\infty$ function:
$$ v = \sum_i f_i e_i$$
Would this all be correct? Incidentally, would the set of all possible $m$ forms be called the span, similar to the vector spaces that we focus on in linear algebra?