Let $M$ be an $n$-dimensional surface, with coordinates $(x^i)$. You can think of $M$ as being $\RR^n$, but we will also consider surfaces in higher-dimensional spaces, such as the 2-sphere in $\RR^3$. Then, as before, $\bigwedge^1(M)$ is the span of the 1-forms $\{dx^i\}$, $\bigwedge^0(M)$ is the space of functions on $M$, and $\bigwedge^p(M)$ is the space of $p$-forms; we will often write simply $\bigwedge^p$ for $\bigwedge^p(M)$.

The basis $\{dx^i\}$ of 1-forms is quite natural, and is called a coordinate basis. However, in the presence of an inner product $g$ on $\bigwedge^1$, it is often preferable to work instead with an orthonormal basis. So let's start over, and work with an arbitrary (for now) basis. Suppose $\{\sigma^i\}$ is a basis of $\bigwedge^1$. Then a basis for $\bigwedge^p$ is \begin{equation} \{\sigma^I\} = \{\sigma^{i_1} \wedge … \wedge \sigma^{i_p}\} \end{equation} where the index set $I = {i_1,…,i_p}$ satisfies $1\le i_1<…<i_p\le n$, and where we are of course assuming $p\le n$.

What about $\bigwedge^0$, the space of scalars, that is functions? The elements of this space contain no factors of $\sigma^i$. There is nonetheless a natural basis for this 1-dimensional space, namely the constant function $1$.