Let's assume $M$ is a smooth $n$-dimensional manifold and $\omega$ is a differential $k$-form on $M$, then i know that $\omega$ can be expressed as $$\omega = \sum_{\mu_1<\cdots<\mu_k}\omega_{\mu_1\cdots \mu_k}dx^{\mu_1}\wedge...\wedge dx^{\mu_k}$$
Where $\omega_{\mu_1\cdots \mu_k} = \omega\left(\frac{\partial}{\partial x^{\mu_1}},...,\frac{\partial}{\partial x^{\mu_k}}\right)$ are the components of $\omega$ w.r.t the canonical basis $\left\{\frac{\partial}{\partial x^{\mu_i}}\right\}$ of $T_pM$.
When I came across the definition of the wedge product, I noticed that I do not know how such differential form $\omega$ evaluates at an arbitrary vector $(v_1,...,v_n) \in T_pM$, i.e. I don't know what
$$\omega(v_1,...,v_n)$$ is supposed to look like.
One of my books states that if $\omega$ is an alternating $k$-form (not a differential form) on the finite dimensional vector space $V$, then $\omega(v_1,...,v_n)$ is the determinant of the metric matrix with entries $\langle v_i,e_i\rangle$ where $e_i$ are the canonical basis vectors. However, on a general smooth manifold, i don't necessarily have an inner product $\langle \cdot,\cdot\rangle$ on $T_pM$.
Question 1:
Assume I'm given a $1$-form $\omega$ on $\mathbb{R}^2$ such as $$\omega = 2xdx - 3ydy,$$ am I correct if I assume that $2x = \omega\left(\frac{\partial}{\partial x}\right)$ and $ -3y = \omega\left(\frac{\partial}{\partial y}\right)$ where $\left\{\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right\}$ is the canonical basis of $T_p\mathbb{R}^2 \cong \mathbb{R}^2$ corresponding to the canonical basis $\{e_1,e_2\}$?
Question 2:
If that's the case, how would $\omega(v)$ for arbitrary (non-basis) element of $T_p\mathbb{R}^2$ look like?
The background for these questions is that when I came across the definition of the wedge product
$\omega\wedge \mu(v_1,...,v_{r+s})$ I noticed that I did not know what $\omega(v_1,...,v_r)$ respectively $\mu(v_{r+1},...,v_{r+s})$ look like.
I hope someone could clarify whether I'm on the right track or where I'm confusing things.