While studying differential geometry of Manifolds,one has surely come across the space of all alternating $k$-tensors on a vector space $V$ (say of finite dimension $n$). We denote it by $$ \Lambda^k(V) := \{T\in {V^{\otimes^k}}: T\text{ is an alternating tensor} \} $$ which has a canonical basis $\{e_{i_1}\wedge e_{i_2}\wedge\cdots\wedge e_{i_k}: 1\leq i_1<i_2<\cdots<i_k\leq n\}$. Now suppose $U$ is an open set in $\mathbb R^n$. Now, a differential $k$ form on $U$ is defined to be a map $$ \omega: U \to \bigcup_{p\in U} \Lambda^k(T_pU) $$ such that the following are satisfied:
$(1)\quad \omega(p)\in \Lambda^k(T_pU)$,
$(2)\quad$ The functions $c_{i_1,\ldots,i_k}: U\to \mathbb R$ are $C^\infty$ smooth for all $1\leq i_1<\cdots<i_k\leq n$, where $$ \omega(p) = \sum_{1\leq i_1<\cdots<i_k\leq n} c_{i_1,i_2,\ldots,i_k}(p) \, dx_{i_1}|_p \wedge \cdots \wedge dx_{i_k}|_p $$ is the unique basis representation of $\omega(p)$ in terms of the canonical basis.
I want to know if there is a specific standard name for the disjoint union $\bigcup_{p\in U}\Lambda^k(T_pU)$ similar to what we have in case of vector fields where we call $TM=\bigcup_{p\in M} T_p(M)$, which is the disjoint union of the tangent spaces, by the name Tangent Bundle and we can think of a vector field as a section of the tangent bundle.
I am curious to know whether we have anything similar for the alternating tensors and whether we can look at the differential forms as a section of the bundle.
Note
I am not familiar with bundles and know very little about them. So, I need some help to get my question answered. Is there anyone who can provide me with a satisfactory one and at the same time give me some insight about what the bundle means for alternating tensors?
