You are correct. You have already split the gift by each of you filing a form for half of the gift.
Definition of a Split Gift: A split gift refers to a gift that is made by a spouse to a third person. The gift is given to a third person for gift tax purposes. It is also known as gift splitting or gift splitting election. ... Under split gift, a married couple can treat gifts made by one spouse as if they were made one-half by each spouse.
