Can a set of surreal numbers be defined with arbitrary cardinality?

It is my understanding that the surreal numbers form a class rather than a set, because their collection is larger than any set. Thus it would seem to follow that for any cardinality, such as $\aleph_n$ or $\beth_n$ for a fixed $n$, a set of surreal numbers can be defined with that cardinality. Is explicitly defining such a set something that can be easily done?

Yes: Since every ordinal is a surreal number, the sets you're looking for can be taken to be the initial ordinals that represent $\aleph_n$, $\beth_n$, and so forth.