I'm struggling with two particular differential forms: the $\flat$ 1-form and the $\natural$ 2-form defined as follows:
Let $X = (X_1,X_2,X_3)$ be a vector field in an open subset $O$ of $\Bbb R^3$. We define the 1-form $X^\flat$ as $$X^\flat(p)[v] = \langle X(p), v\rangle\text{ for any }p\in O\text{ and }v\in\Bbb R^3$$ and the 2-form $X^\natural$ as $$X^\natural(p)[v,w] = \det(X(p), v, w) \text{ for any }p\in O \text{ and }v,w\in\Bbb R^3.$$
They are asking me to study different operations with said operators but I need to express both forms in terms of $X_1,X_2$ and $X_3$ and $dx,dy$ and $dz$ to work with them. Could anyone please help me out in understanding how to do so? Thanks in advance!
EDIT: From what I gathered, I got $$X^\flat(p)[v] = \langle (X_1,X_2,X_3), (v_1,v_2,v_3) \rangle = \sum_{i=1}^3 X_i v_i = \sum_{i=1}^3 X_i dx_i(v) = \left(\sum_{i=1}^3 X_i v_i\right)[v]$$
$$ \begin{aligned} X^\natural(p)[v,w] & = \det\begin{pmatrix} X_1 & dx_1(v) & dx_1(w)\\ X_2 & dx_2(v) & dx_2(w)\\ X_3 & dx_3(v) & dx_3(w)\\ \end{pmatrix} \\ & = X_1(dx_2\wedge dx_3-dx_3\wedge dx_2) \\ & - X_2(dx_3\wedge dx_1-dx_1\wedge dx_3) \\ & + X_3(dx_1\wedge dx_2-dx_2\wedge dx_1) \end{aligned} $$
$$ = 2X_1\;dx_2\wedge dx_3 + 2X_2\;dx_1\wedge dx_3 + 2X_3\;dx_1\wedge dx_2$$
But I can't get to prove certain equalities. I don't know if I have mistaken myself somewhere...