If your trust is a revocable trust then your social security number can be used for any accounts titled in the name of the trust. You will not need a separate tax identification number.
If your trust is an irrevocable trust because you are separating your ownership and your ability to enjoy the funds from yourself, then you will need a separate tax identification number. These sorts of trust are often for insurance proceeds to avoid estate taxes or for family members.
If a married couple creates a revocable trust and one dies, the trust can use the surviving spouse's social security number. If a portion of the trust becomes irrevocable by the terms of the trust agreement, then a new tax identification number will be needed. These types of trusts are commonly called the Decedent's Trust or Credit Shelter Trust.