In Sean Carroll's GR book, the differential $p$-form is defined as a $(0,p)$ tensor that is completely antisymmetric, which I would think is something like $$\frac{1}{2!}(t_{ab}-t_{ba})\textbf {e}^a \otimes \textbf{e}^b.$$ In Zee's GR book, part IX.7, he defined the $p$-form in a similar way, which he writes as $$\frac{1}{2!}t_{ab}dx^a dx^b.$$ Zee also wrote that $dx^a dx^b$ is the wedge product such that $dx^a dx^b=-dx^b dx^a$.
I have troubles reconciling these two definitions. Is $dx^a$ the basis vector $\textbf{e}^a$? How can we start with either one of the expression and manipulate it so that we get the other expression?