Overlapping bookings for the same room

Table bookings:

Find every pair of bookings for the same room whose time ranges overlap. Two bookings overlap when each starts before the other ends; a booking that ends exactly when another starts does not overlap it. List each pair once, with the lower id first.

Output columns: booking_a, booking_b.

CREATE TABLE bookings (
  id int PRIMARY KEY,
  room_id int NOT NULL,
  start_at timestamp NOT NULL,
  end_at timestamp NOT NULL CHECK (end_at > start_at)
);