It seems like you know some differential geometry, so I would recommend you read Loomis and Sternberg's book Advanced Calculus. In appendix II of chapter $11$, they introduce the notion of a vector valued differential form. The definition is as follows:
Let $M$ be a differentiable manifold, and $E$ a Banach space over $\Bbb{R}$. An $E$-valued exterior differential form on $M$, of degree $p$, is a function $\omega$ which assigns to each $x \in M$, a function
\begin{align}
\omega(x): \underbrace{(T_xM) \times \dots \times (T_xM)}_{p \text{ times}} \to E,
\end{align}
whereby $\omega(x)$ is (continuous) multilinear and alternating.
Typically when we discuss differential forms, we take $E= \Bbb{R}$. Hence, we get the notion of a 1-form like $dx,dy,dz$. If instead we only assume that $E$ is a finite-dimensional real vector space, then by choosing a basis $\{e_1, \dots, e_n\}$ for $E$, we can uniquely write every $E$-valued exterior differential form $\omega$ as
\begin{equation}
\omega = \sum_{i=1}^n \omega_ie_i
\end{equation}
where $\omega_i$ are $\Bbb{R}$-valued exterior differential forms (i.e the usual kind which you might be used to).
In such a case, we might write the form $\omega$ in vector notation as $(\omega_1, \dots, \omega_n)$ (again, refer to the book for a slightly more detailed explanation).
A concrete example can be when $E = \Bbb{R}^2$, and we choose the standard basis $e_1,e_2$. Then, once we have the real-valued 1-forms $dx$ and $dy$, we can define the vector-differential form $d\boldsymbol{r}$ as
\begin{equation}
d\boldsymbol{r} := dx \cdot e_1 + dy \cdot e_2 = (dx,dy)
\end{equation}
where the symbols have the meaning as explained above.
Here is the link: https://en.wikipedia.org/wiki/Curvilinear_coordinates#Differential_elements
– Rui Liu Jun 28 '19 at 18:16