There is, yes, using a combination of row_to_json, array_to_json, array_agg and ST_AsGeoJSON. I realize, on rereading your question, that you asked for ogr2ogr approach, but seeing as your source is Postgis, I thought you might appreciate a pure Postgres/Postgis approach. I have used this approach with Google Maps, so it works well if you need to create GeoJSON dynamically, which would be tricker with ogr2ogr.
(SELECT 'FeatureCollection' As type, array_to_json(array_agg(feat)) As features
FROM (SELECT 'Feature' As type,
ST_AsGeoJSON(tbl.geom)::json As geometry,
row_to_json((SELECT l FROM (SELECT id, prop1, prop2, ...) As l)
) As properties
FROM your_table As tbL
WHERE st_intersects(geom, st_setsrid(st_makebox2d(st_makepoint(x1, y1),st_makepoint(x2, y2)), srid))
) As feat
) As featcoll;
where there is an optional where clause with an ST_Intersects. In the inner query, you can add to Select id, prop1, etc, to get a list of any properties that you would like to appear in the output GeoJSON.
The original idea for this comes from this BostonGIS blog
EDIT: If you have Postgres 9.4 or higher, use @dbaston's answer. It is so much cleaner.