As construction began Monday on the elevated Skyway, a groundbreaking agreement could possibly mean higher tolls and better service. The mayor took the first steps towards privatizing the Chicago Skyway this week. City Hall will issue a ''request-for-qualifications'' to private toll operators to gauge interest in the 7.8-mile roadway connecting Chicago and Northwest Indiana. Several dozen bids are expected. The Skyway's $2 toll, last increased in 1993, is likely to be raised, under an agreement that would ''set forth parameters'' for future increases.