If you don't want to enter it, don't. There's your answer.
A more complete answer:
There are several ways to get around the SSN requirement for USA paypal accounts.
Some people don't have an SSN. Could they require someone who doesn't have an SSN to give it to them? No...
Some countries do not require the SSN, for instance, in Canada, it's called a social insurance number (or something) and it's illegal to use that number for any other purpose than what it's made.... but they may ask to provide some other proof, like a passport.
You could set up a business DBA in the name of the paypal account, get a tax ID for that name, and give them that. It's perfectly legal.
You could also upgrade it to a business account, and do the prior step I mentioned.... then contact paypal and say you're just in the accounting department, and don't want to give out your social.
The other option to make a new account with a new name, address, DOB, credit card, bank account, new email, new cookies, new IP address, new LSO files, and start from scratch.
I hope that helps.