If $T \in \wedge^k(V)$ and $S \in \wedge^l(V)$, then the definition of their wedge product is $$T \wedge S := \frac{(k+l)!}{ k ! l!}\text{Alt}(T \otimes S) \in \wedge^{k+l}(V).$$
Meanwhile, we have that $$\text{Alt}(T) = \frac{1}{k!}\sum_{\pi \in S_p} {\operatorname{sgn} \, (\pi)} T^\pi.$$ Now my professor defines for $v_i \in V$
$$v_1\wedge v_2 \wedge \cdots \wedge v_k := \mathrm{Alt} (v_1 \otimes v_2 ...\otimes v_k) .$$
My question: do we need to define $v_1\wedge v_2 \wedge \cdots \wedge v_k$? I think we can derive it by two preceding definitions, but unfortunately I'm getting $ k! \, \operatorname{Alt} (v_1 \otimes v_2 ...\otimes v_k)$. Is that $k!$ I'm getting a miscalculation?