We know that an equivalent definition for orientability for a smooth surface embedded in $\mathbb R^3$ is "to have a normal global vector" (i.e. a Gauss map). Of course every surface is locally orientable (we can choose a normal vector), but are there surfaces which are not orientable?
For example, $\mathbb P^n(\mathbb R)$ is not orientable for $n$ even, but this is not an embedded surface.
Thank you in advance.
A Möbius band is a non-orientable embedded surface.
If you want a maximal non-orientable surface, then I think you need to accept that you'll only get immersions, not embeddings -- for example a Klein bottle or a Boy's surface.