Focusing on exterior powers here is a distraction. The main problem already appears when considering the tensor algebra $T(E)=\oplus_{n\ge 0}E^{\otimes n}$. Once the issue is understood for the tensor algebra, figuring out what to do for the exterior or symmetric algebras (e.g., Fermion or Boson Fock spaces) is trivial, because we are in characteristic zero. In positive characteristic, this becomes subtle as can be seen for example in the recent work "Koszul modules and Green’s conjecture" by Aprodu et al. where a positive characteristic Hermite Reciprocity map is constructed.
Given a vector space $E$, the first step is to consider tensor products like $E^{\otimes n}$. This can be done algebraically as in the mentioned book by Greub. However, when $E$ is an infinite dimensional TVS (topological vector space) the resulting algebraic tensor product $E\otimes\cdots\otimes E$ is a rather unsuitable object for the purposes of analysis. One typically needs to enlarge this space using a completion procedure (topology is essential for that), and one then obtains a topological tensor product $E\widehat{\otimes}\cdots\widehat{\otimes}E$. The caveat is: even when working with Banach spaces, there are lots of ways of doing that. This was Alexander Grothendieck's Ph.D. thesis work. He considered a dozen or so inequivalent reasonable definitions for these completions/versions of the tensor product which depend on the topological structure. In other words, in the course of his explorations Grothendieck found Hell. Luckily, he kept exploring and he eventually also found Paradise: the class of nuclear spaces for which all these different constructions become the same and therefore acquire a cananical feel to them.
Likewise, for the sum $\oplus_{n\ge 0}$ one typically starts with the algebraic direct sum (only finite sums allowed, i.e., we look at almost finite sequences where after a while all the terms are zero) and one then enlarges the space by taking a completion.
The construction by Feldman, Knörrer and Trubowitz is an explicit way (just a choice that works for their purposes) of doing a succession of algebraic constructions followed by topological completions, as explained above.
Now one might think that the algebraic construction as in Greub's book is more general/powerful/etc. than the topological procedure. This is a misconception. For infinite dimensional spaces that are not too big, one could in fact argue the opposite is true. Take for example the simplest infinite dimensional space: $E=\oplus_{n\ge 0}\mathbb{R}$ which can be viewed as the space of almost finite sequences of real numbers, or the space of polynomials in one variable with real coefficients. Then $T(E)$ constructed algebraically à la Greub is a particular case of the topological completion construction. Indeed, equip $E$ with the locally convex topology defined by the set of all seminorms on $E$. This is also called the finest locally convex topology. With this topology, the space is nuclear in the sense of Grothendieck's general definition (but not nuclear in the sense of the more restrictive definition used by the Russian school around Gel'fand et al., namely, the notion of countably Hilbert nuclear spaces). So that is a good sign: pretty much any reasonable completion will give you the same $E\widehat{\otimes}\cdots\widehat{\otimes}E$ which will also coincide with the algebraic tensor product (without hats). Finally for the sum one has several possible choices, but one of them will give the algebraic construction. Let us say that a seminorm on the algebraic direct sum $T(E)$ is admissible if and only if it restricts to a continuous seminorm on each summand. Take the locally convex topology on $T(E)$ defined by the set of all admissible seminorms. Take the completion. This will give nothing new. Note that all seminorms are admissible for the case $E=\oplus_{n\ge 0}\mathbb{R}$ but I wanted to introduce a more general construction which can be applied for example to $E=\mathscr{S}(\mathbb{R})$, the Schwartz space of rapidly decaying smooth functions. Then the $T(E)$ will be isomorphic as a TVS to $\mathscr{D}(\mathbb{R})$, the space of compactly supported smooth functions.
Moral(s) of the story:
For infinite dimensional spaces ordinary bases (Hamel bases) are no good. You need Schauder bases which allow infinite linear combinations. You will need to base your construction on topology. Even when topology seems to be absent, and one uses purely algebraic direct sums and tensor products, topology is still there hiding behind the scenes as in the $E=\oplus_{n\ge 0}\mathbb{R}$ example.
Recommended reading:
The excellent vignette "Schwartz kernel theorems, tensor products, nuclearity" by Paul Garrett.