Can you say "are not we all?" instead of "aren't we all?"
- No, you can't say "are not we all?". It is ungrammatical, as you suggest.
- Yes, you can say either "you need not go there" or "you needn't go there".
The reasons have to do with negative contractions and the fact that they count as a single auxiliary verb. Only one auxiliary verb may be inverted by Subject-Auxiliary Inversion, which is required by Question-Formation. This may be a contracted verb like isn't or don't, inverted with the subject as a unit. If the not is not contracted, on the other hand, then it stays where it is, after the Subject.
Thus these sentences (the "Q" means "Apply Question-Formation")
- We are not all here. ==Q==> Are we not all here?
- We aren't all here. ==Q==> Aren't we all here?
follow the rule for subject-auxiliary inversion, but these don't, and are therefore ungrammatical:
- *Are not we all here?
- *Need not you do that?
In the case of (2), there's no inversion, because it's not a question, so the contraction can be unpacked at will.
Summary: Contractions are only optional in their original position. If they're moved, they're frozen.
Both "are we not all" (or "are not we all") and "you need not go there" are valid here, but that is not the same as saying they are interchangeable with "aren't we all" and "you needn't go there."
In written English, I would probably opt for the written out form, and in some speeches or debates, I might use the full form for rhetorical effect. In modern conversational English, however— certainly in American English— saying are we not or you need not would sound very formal and stilted. The contracted form is overwhelmingly more common, from can't we be friends to wouldn't it be nice. If I heard the long form I would assume someone was quoting from a passage of literature, or making a show of mock pomposity.