I read from a textbook that one defines the wedge product of basis elements as, for example, $$e^{i_1}\wedge...\wedge e^{i_k}\equiv k!\text{Alt}(e^{i_1}\otimes...\otimes e^{i_k})$$ and of two forms $$a^{(p)}\wedge b^{(q)}\equiv \frac{(p+q)!}{p!q!}\text{Alt}(a^{(p)}\otimes b^{(q)})$$ seperately. Intuitively, I am thinking that one needs only one of these definitions and the second will naturally follow. I am trying to realize this manually but annoying factors pop out every time.
Is there a connection between these two definitions? I.e. is there a way to produce one of them as a consequence of the definition of the other?