Typically, in the functor of points approach, one first constructs the category of algebraic spaces by first constructing the category of locally representable sheaves for the global Zariski topology (Schemes) on $CRing^{op}$. That is, taking the full subcategory of $Psh(CRing^{op})$ which consists of objects $S$ such that $S$ is a sheaf in the global Zariski topology and $S$ has a cover by representables in the induced topology on $Psh(CRing^{op})$.  This is the category of schemes.  Then, one takes this category and equips it with the etale topology and repeats to get the category of algebraic spaces.  

*Can we "skip" the category of schemes entirely by putting a different topology on $CRing^{op}$?*

Some motivation: In algebraic geometry, all of our data comes from commutative rings in a functorial way (intentionally vague).  All of the grothendieck topologies with nice notions of descent used in Algebraic geometry can be expressed in terms of commutative rings, e.g., the algebraic and geometric forms of Zariski's Main theorem are equivalent, we can describe etale morphisms in terms of etale ring maps, et cetera. What I'm trying to see is whether or not we can really express all of algebraic geometry as "left-handed commutative algebra + sheaves (including higher sheaves like stacks)".  The functor of points approach for schemes validates this intuition in the simplest case, but does it actually generalize further?

The main question is italicized, but feel free to tell me if I've incorrectly characterized something in the motivation or the background.