I am learning about multilinear maps by myself and the book I'm following gives a definition which is somewhat vague.
That's the definition: Given vector spaces $V_1,V_2,\dots,V_p,W$. A mapping $\phi:V_1\times\dots\times V_p \rightarrow W$ is multilinear if it is linear in each argument.
Let us consider the case in which $p=2$. Let $a \in \mathbb{K}$, $v_1 \in V_1$ and $v_2 \in V_2$. Are the following steps correct?
$$ \phi(av_1,v_2) = a\phi(v_1,v_2)=\phi(v_1,av_2) $$
Yes, they are correct. The first equality holds by the linearity in the first argument, and the second equality holds by the linearity in the second argument.
This is correct and make sure never to confuse 'linear' with 'multilinear'. Multilinear means linear in each of the $p$ variables. $\Lambda^p V$ is the vector space of real-valued $p$-linear alternating forms, meaning that they have to be linear for each of the $p$ variables in exactly the way you just described, as well as being alternating.
I will provide an example which you will be seeing soon anyway in the book if you have not seen it. The wedge (or exterior) product of a $p$-form and a $q$-form is bilinear.
$(\phi_1 + \phi_2) \wedge \psi=\phi_1 \wedge \psi + \phi_2 + \psi,$
$\phi \wedge (a\psi)=a(\phi \wedge \psi).$
