I'm currently thinking about some combinatorics associated to an infinite analogue of the coordinate rings of the Grassmannians $Gr(2,n)$. The combinatorics should be thought of as relating to Plucker coordinates $\Delta^{ij}$ but with $i < j$ arbitrary integers, rather than restricted to $\{1,\ldots ,n\}$. So I've been trying to find the right infinite Grassmannian to have this coordinate ring (or, if the regular functions are a bit more complicated in the infinite case, to at least have these Plucker coordinates in there). I've looked (briefly) into:

(a) Kac's construction of infinite Grassmannians ([Kac, Infinite dimensional Lie algebras, 3rd ed.], Exercise 14.32, p.339)

(b) taking the union of the finite Grassmannians to get a classifying space for $O(n)$ or $U(n)$

(c) infinite Grassmannians coming from Hilbert spaces ([Pressley and Segal, Loop groups])

but none of these seem to quite describe what I want. Some of these are working with $\mathbb{N}$-dimensional space rather than $\mathbb{Z}$-dimensional space (i.e. something more like $\mathbb{C}[t]$ than $\mathbb{C}[t,t^{-1}]$), usually from a colimit of the finite ones, and I can't see how to alter the definition and be sure of keeping the theorems. And with the others that do work with something like $\mathbb{C}[t,t^{-1}]$, I can't see a description that corresponds to planes in that space (and I definitely need just the planes).

I feel sure this is well-known so does anybody know a reference for both the construction I want and also enough information about its coordinate ring?

Edit: Having thought about this a little more, I want to formulate the question more specifically as:

Let $V=\mathbb{C}[t,t^{-1}]$ and define $Gr(2,V)$ to be the set of 2-dimensional subspaces of $V$. Does the finite-dimensional machinery of the Plucker embedding work in this setting and give Plucker coordinates in $\mathbb{C}[Gr(2,V)]$ of the form $\Delta^{ij}$ for integers $i < j$?