A trust doesn't give out SSNs. At best, a trust will provide it's own EIN when required, to a recipient of funds or other property from that trust.
As an addendum to Lisa's response, there are many different types of trusts, and just as many ways each type can be set up. A majority of trusts filing requirements depends on the laws of your state that governs trusts too. So if you're unsure, you really should seek professional help.
When a trust or estate files a tax return that results in fines and penalties because of an error, incorrect filing, or being late, those fines and penalties are not paid by the trust or estate. They come out of the wallet of the legally recognized/appointed administrator. Many times the costs can be so high as to make the cost of professional help seem like a pittance in comparison.
So if this is a continuing trust, getting professional help in at least the first year can save the administrator grief and money in the long run.