$V, W$ vector fields, $\omega$ one-form, what is the definition of $d\omega(V, W)$?
I have seen some equalities and I suppose that $d\omega(V,W)=(V^jW^i-V^iW^j) \frac{\partial f_i}{\partial x^j}$ where $\omega=f_i dx^i$ and $V=V^i \frac{\partial}{\partial x^i}$, $V:M \to \mathbb{R^n}, V^i : M \to \mathbb{R}$ but I would like to confirm.
It is a $2$-form defined as $$d\omega(X,Y)=X(\omega(Y))-Y(\omega(X))-\omega([X,Y])$$ sometimes there is a factor $\frac{1}{2}$ in the right hand side. Moreover $$d(f_ida_i)=df_i\wedge da_i + (-1)^{0}f_id^2a_i=df_i\wedge da_i $$
Example
If $\omega=Pdx+Qdy,$ then
$$d\omega=dP\land dx+dQ\land dy \\=(\frac{\partial P}{\partial x}dx+\frac{\partial P}{\partial y}dy)\land dx+(\frac{\partial Q}{\partial x}dx+\frac{\partial Q}{\partial x}dy)\land dy \\=\frac{\partial P}{\partial y}dy\land dx+\frac{\partial Q}{\partial x}dx\land dy \\=(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dx\land dy$$ Note that $dx\land dy=-dy\land dx$