Is it possible to define schemes without using sheaves?

Question

My understanding is that smooth manifolds can be defined either in the usual way (using smooth charts and atlases), or using sheaves (as a locally ringed space which is locally isomorphic to the sheaf of smooth functions on $\mathbb{R}^{n}$). However, schemes seem to only be defined using sheaves. Is it possible to define a scheme as a topological space which is locally homeomorphic to an affine scheme, such that the transition maps are all maps of affine schemes? Another way to phrase the question could be: given such a space, is it possible to recover a structure sheaf defining a scheme?