I'm following Theodore Shifrin's course Math 3510 on youtube. So I will use his notations.
If $\omega \in \Lambda^k(\mathbb{R}^n)^*$ is an alternating multilinear map (i.e., a $k$-form on $\mathbb{R}^n$) I visualize its geometrical meaning. Moreover since a $k$-form is a function, I know how to evaluate it on $k$ input vectors. For example if $\omega = 3dx \wedge dy + 5dx \wedge dz + 7dy \wedge dz = 3dx_{12} + 5dx_{13} + 7dx_{23}$ is a 2-form on $\mathbb{R}^3$ and $v = \begin{pmatrix} 1 \\ 3 \\ 4\end{pmatrix}$ and $u = \begin{pmatrix} 5 \\ 2 \\ 11\end{pmatrix}$ then
$$ \omega(v,u) = 3\begin{vmatrix} 1 & 5 \\ 3 & 2 \end{vmatrix} + 5\begin{vmatrix} 1 & 5 \\ 4 & 11 \end{vmatrix} + 7\begin{vmatrix} 3 & 2 \\ 4 & 11 \end{vmatrix} = 91 $$
Let $\mathcal{C}^\infty(\mathbb{R}^n)$ be the ring of smooth scalar functions from $\mathbb{R}^n$ to $\mathbb{R}$. At the beginning of this lecture prof. Shifrin defines the set $\mathcal{A}^k(\mathbb{R}^n)$ of differential $k$-forms on $\mathbb{R}^n$ as the module over the ring $\mathcal{C}^\infty(\mathbb{R}^n)$ generated by $\Lambda^k(\mathbb{R}^n)^*$. An element of $\mathcal{A}^k(\mathbb{R}^n)$ is a sum $\eta = \sum_{I} f_Idx_I$ where each $I$ in the summation is a multi-index of cardinality $k$.
How should I think about $\eta$ ? Can I look at $\eta$ as a function that takes vectors as inputs and returns a real number?
You should think of it as taking $k$ smooth vector fields as inputs and returning a smooth function. So, if $X_1,\dots,X_k$ are vector fields, you'll have $$\big(\eta(X_1,\dots,X_k)\big)(x) = \eta(x)(X_1(x),\dots,X_k(x)).$$ In particular, if $\eta = \sum f_I\,d\mathbf x_I$, you get the value $\sum f_I(x)\,d\mathbf x_I(X_1(x),\dots,X_k(x))$.