Relationship between differential forms and coordinates For the purpose of this question let us restrict our considerations to smooth $3$-manifolds. So the manifold $M$ we consider here is endowed with smooth coordinate charts $(x,y,z)$. 
What I have gathered so far about differential forms:
If $x$ is first coordinate of a chart then we use $dx$ to denote a differential one form. This notation is not just notation: the differential one form is in fact the exterior derivative of the first coordinate of the chart. 
Now let's move to the $n$-dimensional case for my actual question:

Does it mean that on an $n$-manifold the differential $1$-forms are
  always $dx_1, \dots, dx_n$?

Elaboration of my question: 
If the answer is yes it would mean that whenever I'm trying to find all possible $1$-forms on say, the torus, then I don't have to think: Since the torus is $2$-dimensional we choose to denote its coordinate charts by $(x,y)$ and then it's clear that the differential one forms are $dx, dy$ (and all possible linear combinations thereof with coefficients some smooth functions). 
 A: On a smooth manifold with a smooth chart $(U, \phi)$, the $1$-forms $dx^1, \ldots, dx^n$ form a local coframe of $TM$ defined on $U$, that is, for any $1$-form $\alpha$ on $M$, there are smooth functions $\alpha_i$ such that $\alpha\vert_U = \alpha_1 dx^1 + \cdots + \alpha_n dx^n$; this only requires knowing that for an $n$-manifold the rank of $TM$ is $n$, a characterization of smoothness, and that the coordinate $1$-forms are linearly independent. Better yet, for any closed $1$-form $\beta$ on $M$ (that is, a $1$-form $\beta$ such that $d \beta = 0$) and any point $p \in M$ such that $\beta_p \neq 0_p$, there is a neighborhood $V$ of $p$ and coordinates $(x^i)$ on $V$ so that $\beta = dx^1$.
However, in general there is no guarantee that one can do this globally. For example, consider the $1$-form on $\mathbb{R}^2 - \{0\}$ defined by $$\gamma := -\frac{y}{x^2 + y^2} dx + \frac{x}{x^2 + y^2} dy,$$ which direct computation shows is closed.
We can readily describe coordinates in which this is a coordinate $1$-form In any chart $W$ with polar coordinates $(r, \theta)$, the restriction $\gamma\vert_W$ coincides with $d\theta$, that is, this $1$-form measures the infinitesimal change in angle about the origin of any vector it is evaluated on. However, there is no global chart for which $\gamma$ is a coordinate $1$-form:
Consider any closed path $\zeta$ in $\mathbb{R}^2 - \{0\}$ that winds around the origin once counterclockwise (e.g., the unit circle with the standard parameterization). Then, by Stokes' Theorem, for any global coordinates $(x^i)$, we have
$$\int_{\zeta} dx^i = \int_{\partial \zeta} x^i = \int_{\emptyset} x^i = 0,$$
but using the angular characterization of $\gamma$ above,
$$\int_{\zeta} \gamma = 2 \pi.$$
(For this reason, the common practice of writing $\gamma$ as $d \theta$ even globally is misleading.)
