I'm reading "Elements of Noncommutative Geometry" by Garcia-Bondía. There it was mentioned that the algebra $\Gamma^\infty(\mathbb{C}l(M))$ is generated by $\Omega^1(M)$.
Here $\Omega^1(M)$ are the 1-forms on the manifold $M$. The complex Clifford bundle is given by $\mathbb{C}l(M)\to M$ with $\mathbb{C}l(M):=\mathbb{C}l$ $T^*M$ and $\Gamma^\infty(\cdot)$ is the set of smooth sections of a fibre bundle.
In this regard, $\alpha\in\Omega^1(M)$ maps $M$ to $T^*M$ s.t. $\alpha(p)\in T^*_pM$. On the other hand $\phi\in\Gamma^\infty(\mathbb{C}l(M))$ is a smooth map from $M$ to $\mathbb{C}l(M)$ satisfying $\phi(p)\in\mathbb{C}l(p)=\mathbb{C}l$ $T^*_pM$. Constructing a Clifford algebra, I need a quadratic form. Therefore, I do not know how to continue.
How can I show that the 1-forms generate the algebra?
Thanks for your help.