Is it a crime to give a fake Social Security number?

In the United States, the only time someone will ask you for your SSN it is for something important (like applying for job to prove that you are authorized to work, or applying for a financial relationship with an institution in form of credit card/loan or signing up for a service with recurring payments, like electricity or cable).  While you may not realize it, immediately thereafter, you might also be signing your name or at least verbally agreeing to the terms, which are in form of a legal contract.  Thus, if you give a false SS#, you are essentially lying (committing perjury), and that has serious legal consequences.  In case of a business, things can get pretty bad, for instance, you might lose the privilege the business extended to you, but they can also sue you, but when you are in front of a Government agent of any kind and you provide a false name or identifying information like SS#, you have chosen to put yourself in a lot of legal trouble.