Wikipedia's intrinsic definition of differential forms query The “Intrinsic definitions” section of the Wikipedia article on differential forms here states:

By the universal property of exterior powers, this is equivalently an
alternating multilinear map:
$$\beta_{p}:\bigoplus_{n=1}^{k}T_{p}M\rightarrow\mathbb{R}.$$

Where $\beta_{p}$ is a $k$-form at point $p$. Can anyone provide a not too technical explanation as to what that means? From Wikipedia I sort of understand that $\oplus$ is a direct sum (eg $\mathbb{R}\oplus\mathbb{R}$ gives $\mathbb{R}^{2}$, the Cartesian plane). But what does the direct sum of tangent spaces mean? And how does this give an alternating multilinear map? Is it more or less saying that a $k$-form acts on $k$ tangent vectors to give a real number?
 A: Let $M$ be a smooth manifold. A differential $k$-form on $M$ is a smooth section of the bundle $\varpi : \bigwedge^kT^*M \to M$, i.e. a map $\beta : M \to \bigwedge^kT^*M$, $p \mapsto \beta_p$ such that $\varpi\circ\beta = \operatorname{id}_M$. In particular, $\beta_p \in \bigwedge^kT_p^*M$ for every $p$. So a differential $k$-form on $M$ can be thought of as a collection $\{\beta_p\}_{p \in M}$ where $\beta_p$ varies smoothly as $p$ varies.
To understand what $\beta_p$ is, note that under the isomorphism $\bigwedge^k T_p^*M \cong \left(\bigwedge^kT_pM\right)^*$, the element $\beta_p$ of the left hand side corresponds to an element of the right hand side (which I will also denote by $\beta_p$), a linear map $\beta_p : \bigwedge^kT_pM \to \mathbb{R}$.
There is a natural multilinear map $\pi : \bigoplus_{i=1}^kT_pM \to \bigwedge^k T_pM$ given by $(v_1, \dots, v_k) \mapsto v_1\wedge\dots\wedge v_k$. Precomposing $\beta_p$ with this map, we obtain a multilinear map $\beta_p\circ\pi : \bigoplus_{i=1}^kT_pM \to \mathbb{R}$. Note that if $v_i = v_j$ for some $i \neq j$, then $\pi(v_1, \dots, v_k) = 0$ and hence $(\beta_p\circ\pi)(v_1, \dots, v_k) = 0$; that is, $\beta_p\circ\pi : \bigoplus_{i=1}^kT_pM \to \mathbb{R}$ is an alternating multilinear map.
The correspondence between linear maps $\bigwedge^kT_pM \to \mathbb{R}$ and alternating multilinear maps $\bigoplus_{i=1}^k T_pM \to \mathbb{R}$ given by precomposing with $\pi$ is bijective. In particular, for every alternating multilinear map $\hat{L} : \bigoplus_{i=1}^kT_pM \to \mathbb{R}$, there is a unique linear map $L : \bigwedge^kT_pM \to \mathbb{R}$ such that $\hat{L} = L\circ\pi$. The map $L$ is constructed by defining $L(v_1\wedge\dots\wedge v_k) := \hat{L}(v_1, \dots, v_k)$ and extending linearly.
What the previous paragraph tells us is that the map $\beta_p\circ\pi : \bigoplus_{i=1}^kT_pM \to \mathbb{R}$ is uniquely determined by $\beta_p$. This is why Wikipedia also uses $\beta_p$ to denote the map which I have called $\beta_p\circ\pi$.
