I have been reading on the wedge product (From Shutz's Geometrical Methods of Mathematical Physics) and I don't quite get why the wedge product is associative.
The book defines the wedge product of two $1$-forms as:
$$p\wedge q:=p\otimes q-q\otimes p.$$
Now writing down $p\wedge (q\wedge r)$ I get:
$$\begin{align} p\wedge (q\wedge r) &=p\otimes (q\wedge r)-(q\wedge r)\otimes p\\ &=p\otimes (q\otimes r - r\otimes q)-(q\otimes r - r\otimes q)\otimes p\\ &=p\otimes q \otimes r - p\otimes r \otimes q - q\otimes r\otimes p + r\otimes q \otimes p. \end{align} $$
The latter is different from what I've computed doing it the other way, and different from what I've found online, even more, the expression for the wedge product of three forms has six terms everywhere I've looked.
Am I misunderstanding something here?
Thanks in advance.