I'm studying differential forms and some linear algebra doubts popped up:
Given $V$ a vector space, we define $A_k(V)$ as the space of alternating $k$-linear maps $V^k \to \mathbb{R}$. The tensor product between to maps $f \in A_k(V)$ and $g \in A_l(V)$ is defined by $(f \otimes g)(v_1, \dots, v_{k+l}) := f(v_1, \dots, v_k)g(v_{k+1}, \dots, v_{k+l})$. Then, we define the wedge product between $f$ and $g$ as $$f \wedge g := \sum_{\sigma \in S_k} (\text{sgn } \sigma)\sigma(f \otimes g) = \sum_{\sigma \in S_k} (\text{sgn } \sigma)f(v_{\sigma(1)}, \dots, v_{\sigma(k)})g(v_{\sigma(k+1)}, \dots, v_{\sigma(k+l)}).$$
Now, for another definition: given two vector spaces $V, W$, we define the tensor product of $v \in V$ and $w \in W$ by $(v \otimes w)(f,g) = f(v)g(w)$, where $f \in V^*$ and $g \in W^*$. Then we define the wedge product of $v_1, \dots, v_k \in V$ as $$v_1 \wedge \cdots \wedge v_2 := \sum_{\sigma \in S_k} (\text{sgn } \sigma) v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(k)}.$$
The $k$-wedge product of $V$, denoted $\wedge^k V$, is the span of these wedge products in vectors. Which is the relation between these two definitions? I'm trying to undersand $k$-forms and in this setting, wedge products are really important to define a basis of the differentials.