Being under law means being under it's authority.
Paul is saying in verse 19 that the law, the Ten Commandments was to stop the mouth of every sinner that they will know that they are all guilty before God without Christ, for by the Law he said in verse 20 is the knowledge of SIN. In the book of 1Jn3:4 the apostle John supports Paul by saying.
Whosoever committeth SIN transgresseth also the Law: for SIN is the transgression of the Law. The obvious question now is, is there sin in the world? And the equally obvious answer is YES. Since SIN is still in this world then God’s Ten Commandments must be still binding upon us.
Paul says in Rom 3:31 Do we then make void the Law through faith? God forbid; yea we establish the law. There are no contradictions here beloved, the Ten Commandments are still binding on us today, Both apostles Paul & John explains clearly that its purpose is to identify SIN, Paul further makes it clear that we cannot make void the Ten Commandments, GOD forbids it Rom8:31.