Given a tangent space $T_xM$, where $M$ is a differentiable manifold homeomorphic to $\mathbb{R}^n$, we have the cotangent space $T^{*}_xM$ defined as being the set of linear functionals $\eta: T_xM \to \mathbb{R}$. Given the equivalence class definition of a tangent vector that is, $[x,i,a] \in T_xM$ where $[x,i,a] \in M \times \Lambda \times \mathbb{R}^n$. where $\Lambda$ is the index set of the charts. Let $\pi_M: T_xM \to M$
My questions are as follows:
How do we construct such an $\eta$? Is it simply $$f \circ \psi_j \circ \pi_M: T_xM \to M \to \mathbb{R}^n \to \mathbb{R}?$$ or is it $$f \circ \pi_M: T_xM \to \mathbb{R}^n \to \mathbb{R}?$$where $\psi_j$ is a map in the atlas on $M$ and $f$ is any function $f : \mathbb{R}^n \to \mathbb{R}$.
Moreover, why is the cotangent bundle defined as $T^{*}M = \{(x,\eta): x \in M, \eta \in T^{*}_xM \}$ defined the way it is? Why is it not defined as the set of all $\eta$?