Exterior derivarive dependent only on point For any one-form (a linear form on the tangent space of each point) we have its exterior derivative $d\omega$ which is a two-form defined by $d\omega(X,Y)=D_X(\omega(Y))-D_Y(\omega(X))-\omega([X,Y])$ 
I'm trying to show that the value of $dw(X,Y)$ at a point $p$ is depndent only on the values of the vetcor fields $X,Y$ at that point.
I've just started diving into this subject, and I really don't have a good direction. I tried to think of all the tools I have so I thought of decomposing $X,Y$ to some orthonoraml base, or maybe even decompose $\omega$ by $\sum_i\omega(X_i)\omega^i$ where $\omega^i$ is the linear form associated with $X_i$ i.e for any $i,j$: $\omega^i(X_j)=\delta_{ij}$
I will be very glad for some help! 
In addition, any general insight on this new definition of exterior derivative can be helpful. For now it seems quite peculiar.
Thanks!
 A: First, we should define tensors. There are various types of tensors, and we focus on those which are relevant to your question. A tensor of type $(k,0)$ is a map$$T:\Gamma(TM)\times\ldots\times\Gamma(TM)\to C^\infty(M),$$which is $C^\infty(M)$-linear in every coordinate. In other words, $T$ eats $k$ vector fields and spits out a function. The linearity means that $T$ is additive in each coordinate, and for vector fields $X_1,\ldots,X_k$ and a function $f\in C^\infty(M)$ we have$$T(fX_1,X_2,\ldots,X_k)=\ldots=T(X_1,\ldots,X_{k-1},fX_k)=fT(X_1,\ldots,X_k).$$
The following is a fundamental observation about tensors:
Proposition: Let $T$ be a $(k,0)$ tensor. Then the value of $T(X_1,\ldots,X_k)$ at a point $p\in M$ depends only on the values of $X_1,\ldots,X_k$ at $p$. In other words, if $Y_1,\ldots,Y_k$ are also vector fields and we have $X_i(p)=Y_i(p),\quad i=1,\ldots k,$ then $T(X_1,\ldots,X_k)(p)=T(Y_1,\ldots,Y_k)(p)$.
Proofs of the proposition can be found in many textbooks. Specifically, I guess it is proved in doCarmo's Riemannian Geometry and in Lee's Smooth Manifolds.
To solve your question, you now need to show that $d\omega$ is a tensor. Namely, you should show $C^\infty(M)$-linearity. This is not so hard. You need Leibniz rule, and you need to know the expression for $[fX,Y]$, where $X$ and $Y$ are vector fields and $f$ is a function.
