I'm a little bit confused about the exterior algebra of smooth differential forms $\Omega(M)$ on a manifold M. The definition of k-forms is clear to me, but I don't understand how to put them together, s.t. they form $\Omega(M)$ so to speak. Maybe you can help me to get rid of my problems:
First of all there is some confusion about the definition of $\Omega(M)$. Some people write $\Omega(M):=\bigoplus\limits_{k=0}^{\dim(M)}\Omega^k(M)$ and some $\Omega(M):=\sum\limits_{k=0}^{\dim(M)}\Omega^k(M)$. I never saw the second notation before, are they both the same by definition or is there a different meaning by the second one?
Furthermore, if we accept the definition $\Omega(M):=\bigoplus\limits_{k=0}^{\dim(M)}\Omega^k(M)$ then the elements of $\Omega(M)$ will consist of tuple like $(\omega_0,...,\omega_{\dim(M)})$, whereas $\omega_k\in\Omega^k(M)$. How do I extend the definition of the wedge product of single forms, i.e. $w_i\wedge\omega_j$, to elemts of the algebra, i.e. $(\omega_0,...,\omega_{\dim(M)})\wedge (\alpha_0,...,\alpha_{\dim(M)})=?$ I suggest that one writes the elements as "formal sums" like $(\omega_0,...,\omega_{\dim(M)})=:\omega_0+...+\omega_{dim(M)}$ and then extend the wedge product bilinearly. Is that right?
I hope someone can help me by answering my two questions.
Regards