Field and dimension of the set of all differential forms (as a vectorspace)

Question

Wikipedia says

The set of all differential $k$-forms on a manifold $M$ is a vector space, often denoted $\Omega^k(M)$.

Over which field is this vector space defined?

Let $\dim(M)=:n$. What is the dimension of the vector space $\Omega^k(M)$?