I am reading Tu's book on Manifolds. As I understand it, given a point $p$ of a manifold $M$ we have the basis $\{\frac{\partial}{\partial x^i}|_p\}$ for $T_pM$, where $(U,x^1,...,x^m)$ is a chart around $p$. Now, if I prove that $\{ dx^i(p) \}$ is a basis for $T^\ast_p M$, then it shouldn't be too hard to show that $\{ dx^{i_1}(p)\wedge ... \wedge dx^{i_k}(p) \}$ is a basis for $ \bigwedge^k T^*_pM $.
Now, I think that all I need to do to prove that $\{ dx^i(p) \}$ is a basis for $T^\ast_p M$ is to show that $dx^i(p)(\frac{\partial}{\partial x^j}|_p)=\delta^i_j$.
Let $f\in C^\infty(\mathbb{R})$. We have $$ dx^i(p)(\frac{\partial}{\partial x^j}|_p)(f) =\frac{\partial}{\partial x^j}|_p(f\circ x^i) $$ and this does not seem to be $\delta^i_jf$.