Determinant and wedge product confusion

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?

• It's just a matter of convention, though the first expression seems to be more usual – user8268 Jun 19 at 17:46
• But doesn't that $\frac{1}{n!}$ make the two expressions different? – Peter4075 Jun 19 at 17:53
• it does. It's just a different convention of what the LHS means – user8268 Jun 19 at 19:27
• Some authors choose to have the volume of the unit cube be $1/n!$ to simplify a few other formulas. Others of us would rather have the cube have volume $1$. – Ted Shifrin Jun 20 at 17:39
• There is nothing about wedge products on page 210 of either of the published editions of Introduction to Smooth Manifolds. You must be using one of the pirated draft versions of the first edition, which somebody posted illegally on the internet. Those are full of mistakes and come with no guarantees. If you look at the published second edition, you'll find this discrepancy explained on pages 356-358. – Jack Lee Jun 20 at 20:57