Definition of Wedge Product on Exterior Algebra

Question

I'm confused on a detail of the construction of the exterior algebra $\Lambda(V^*)$ from the individual exterior powers $\Lambda^k(V^*)$ of the dual of an $n$-dimensional vector space $V$. We define it as the vector space $$\Lambda(V^*) = \bigoplus_{k=0}^n \Lambda^k(V^*),$$ and then the most important observation is that the wedge product turns this vector space into an associative, anti-commutative graded algebra. But at this point in the construction the wedge product is only defined as an operator that takes an element of $\Lambda^k(V^*)$ and an element of $\Lambda^\ell(V^*)$ as inputs---it doesn't yet know how to act on elements of $\Lambda(V^*)$, which are $n+1$-tuples of elements from each exterior power. How do we extend the definition of the wedge product to $\Lambda(V^*)$?