Yes you can but it is difficult.
Foreign mortgages in US have gotten more difficult to find since 9/11. Few of the banks are keen.
If it is a new property, talk to the developer and he will refer you to the lender(s) he is using for his US buyers; if they have approved the property to mortgage against, they may come up a with a special deal for you. They treat you as if you have no financial history - a no documentation loan - and you get pretty punitive rates as a result. Expect to pay 6.75%-7.5% and above.
The less you borrow of the value, the better the rate.
Re. the number you need, you need an ITIN (International Tax ID Number), not an SS number. A US accountant can get one for you for about €75.