https://en.wikipedia.org/wiki/Symmetric_algebra
If I understand that article correctly, the symmetric algebra $S(\mathbb R^n)$ is (isomorphic to) the algebra of polynomials with $n$ variables. As a vector space, it is infinite-dimensional, despite finite $n$ (consider $\{1,x,x^2,x^3,x^4,\cdots\}$ as basis vectors). It has a multiplication that is commutative and associative.
Can this be described in terms of the exterior algebra $\Lambda(\mathbb R^\infty)$?
(By $\mathbb R^\infty$ I mean $\mathbb R^\mathbb N$, the space of Real-valued sequences... or perhaps $\ell^\infty$; I'm not familiar enough with these to know what "ought" to be used.)
For $n=2$, consider these infinite sums of bivectors, where $e_ie_j=e_i\wedge e_j$ is the exterior product, and all $e_i$ and $f_i$ are linearly independent:
$$x = e_1e_2+e_3e_4+e_5e_6+\cdots$$ $$y = f_1f_2+f_3f_4+f_5f_6+\cdots$$
The exterior product is associative, and anticommutative for vectors, but commutative for bivectors: $(e_1e_2)(e_3e_4) = -e_1e_3e_2e_4 = e_1e_3e_4e_2 = -e_3e_1e_4e_2 = (e_3e_4)(e_1e_2)$. So we can write the powers of $x$ as $$x^2 = (e_1e_2)(e_1e_2)+(e_1e_2)(e_3e_4)+(e_3e_4)(e_1e_2)+(e_3e_4)(e_3e_4)+\cdots$$ $$= 0+2(e_1e_2e_3e_4)+2(e_1e_2e_5e_6)+2(e_3e_4e_5e_6)+\cdots$$ $$x^3 = 6(e_1e_2e_3e_4e_5e_6)+6(e_1e_2e_3e_4e_7e_8)+\cdots$$ $$x^4 = 24(e_1e_2e_3e_4e_5e_6e_7e_8)+\cdots$$
and the product $xy$ as
$$xy = (e_1e_2f_1f_2)+(e_1e_2f_3f_4)+(e_3e_4f_1f_2)+(e_3e_4f_3f_4)+\cdots$$ $$= yx$$
and so on.
It should be clear that all of these terms $\{1,x,y,x^2,xy,y^2,x^3,x^2y,\cdots\}\subset\Lambda(\mathbb R^\infty)$ are linearly independent, as they are in the ordinary polynomial space. So, is the algebra generated by these bivectors isomorphic to the 2-variable polynomial algebra $S(\mathbb R^2)$?
Are there problems with infinity, or anything else I'm missing?