Determinant and wedge product confusion

Question

Beginner with differential forms – please go easy. I'm trying to understand how the wedge product can be used to define the determinant. In Lee's Introduction to Smooth Manifolds (bottom of page 210) he gives$$\omega^{1}\wedge\cdots\,\wedge\omega^{k}\left(X_{1},\ldots,X_{k}\right)=\det\left(\omega^{i}\left(X_{j}\right)\right).$$However, in this question Can one define wedge products using determinants for $n$-forms? one of the comments gives$$\varphi_{1}\wedge\cdots\wedge\varphi_{n}(v_{1},\dots,v_{n})=\frac{1}{n!}\det(\varphi_{i}(v_{j})).$$I might have the wrong end of the stick, but both things look the same except for that $\frac{1}{n!}$. What have I missed?