Selecting a JSON object of arrays from a PostgreSQL table

Hello and happy new year!

I have prepared a simple SQL Fiddle demonstrating my problem:
http://sqlfiddle.com/#!17/2c9c5/1

In a two-player game I store user chats in a table:

CREATE TABLE chat(
gid integer, /* game id */
uid integer, /* user id */
created timestamptz,
msg text
);

Here I fill the table with a simple test data:

INSERT INTO chat(gid, uid, created, msg) VALUES
(10, 1, NOW() + interval '1 min', 'msg 1'),
(10, 2, NOW() + interval '2 min', 'msg 2'),
(10, 1, NOW() + interval '3 min', 'msg 3'),
(10, 2, NOW() + interval '4 min', 'msg 4'),
(10, 1, NOW() + interval '5 min', 'msg 5'),
(10, 2, NOW() + interval '6 min', 'msg 6'),
(20, 3, NOW() + interval '7 min', 'msg 7'),
(20, 4, NOW() + interval '8 min', 'msg 8'),
(20, 4, NOW() + interval '9 min', 'msg 9');

And can fetch the data by running the SELECT query:

SELECT ARRAY_TO_JSON(
COALESCE(ARRAY_AGG(ROW_TO_JSON(x)),
array[]::json[])) FROM (
SELECT
gid,
uid,
EXTRACT(EPOCH FROM created)::int AS created,
msg
FROM chat) x;

which returns me a JSON-array:

[{"gid":10,"uid":1,"created":1514813043,"msg":"msg 1"},
{"gid":10,"uid":2,"created":1514813103,"msg":"msg 2"},
{"gid":10,"uid":1,"created":1514813163,"msg":"msg 3"},
{"gid":10,"uid":2,"created":1514813223,"msg":"msg 4"},
{"gid":10,"uid":1,"created":1514813283,"msg":"msg 5"},
{"gid":10,"uid":2,"created":1514813343,"msg":"msg 6"},
{"gid":20,"uid":3,"created":1514813403,"msg":"msg 7"},
{"gid":20,"uid":4,"created":1514813463,"msg":"msg 8"},
{"gid":20,"uid":4,"created":1514813523,"msg":"msg 9"}]

However I would like to use "gid" as JSON object properties and the rest
data as values in that object:

{"10": [{"uid":1,"created":1514813043,"msg":"msg 1"},
{"uid":2,"created":1514813103,"msg":"msg 2"},
{"uid":1,"created":1514813163,"msg":"msg 3"},
{"uid":2,"created":1514813223,"msg":"msg 4"},
{"uid":1,"created":1514813283,"msg":"msg 5"},
{"uid":2,"created":1514813343,"msg":"msg 6"}],

"20": [{"uid":3,"created":1514813403,"msg":"msg 7"},
{"uid":4,"created":1514813463,"msg":"msg 8"},
{"uid":4,"created":1514813523,"msg":"msg 9"}]}

Is that please doable by using the PostgreSQL JSON functions?

Thank you
Alex