Is there any standard for JSON API response format?

Yes there are a couple of standards (albeit some liberties on the definition of standard) that have emerged:

1. JSON API - JSON API covers creating and updating resources as well, not just responses.
2. JSend - Simple and probably what you are already doing.
3. OData JSON Protocol - Very complicated.
4. 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.

Google JSON guide

Success response return data

{
  "data": {
    "id": 1001,
    "name": "Wing"
  }
}

Error response return error

{
  "error": {
    "code": 404,
    "message": "ID not found"
  }
}

and if your client is JS, you can use if ("error" in response) {} to check if there is an error.

I guess a defacto standard has not really emerged (and may never). But regardless, here is my take:

Successful request:

{
  "status": "success",
  "data": {
    /* Application-specific data would go here. */
  },
  "message": null /* Or optional success message */
}

Failed request:

{
  "status": "error",
  "data": null, /* or optional error payload */
  "message": "Error xyz has occurred"
}

Advantage: Same top-level elements in both success and error cases

Disadvantage: No error code, but if you want, you can either change the status to be a (success or failure) code, -or- you can add another top-level item named "code".