Why is there a one-to-one correspondence between vector fields and differential 1-forms?

Question

I did not understand the highlighted text. Could anyone please explain it to me. There is a related post here- Differential Forms and Vector Fields correspondence. The first paragraph of the first answer talks about the same thing but I did not understand it either.

Answer 1

Don't make things too hard on yourself. If $\mathbf F(\mathbf x) = (F_1(\mathbf x),\dots,F_n(\mathbf x))$ is a vector field on $\Bbb R^n$, the corresponding $1$-form is $\omega = F_1(\mathbf x)dx_1+\dots+F_n(\mathbf x)dx_n$. Note that for any $\mathbf v\in\Bbb R^n$, we have $\langle\mathbf F(\mathbf x),\mathbf v\rangle = \omega(\mathbf x)(\mathbf v)$.