Here goes a possibly simple question, but since I don't know the answer...
Consider a manifold $M$ and the differential graded algebra of forms on it:
\begin{equation} \Lambda(M) = \bigoplus_{k=0}^{n}\Lambda^{k}(M) \end{equation}
with $\Lambda^{0}(M)$ being the algebra of smooth functions on $M$. My question: is $\Lambda(M)$ generated by $\Lambda^{0}(M)$ and $\Lambda^{1}(M)$, in the sense that every form $\omega \in \Lambda(M)$ is equal to a sum of wedge products of elements of $\Lambda^{0}(M)$ and $\Lambda^{1}(M)$ ? If so, how can I prove it? If not, to which extent is something of that sort true?
Thanks!
[edited for typos]