Yes there are a couple of standards (albeit some liberties on the definition of standard) that have emerged:
- JSON API – JSON API covers creating and updating resources as well, not just responses.
- JSend – Simple and probably what you are already doing.
- OData JSON Protocol – Very complicated.
- HAL – Like OData but aiming to be HATEOAS like.
There are also JSON API description formats:
- Swagger
- JSON Schema (used by swagger but you could use it stand alone)
- WADL in JSON
- RAML
- HAL because HATEOAS in theory is self describing.