The state religious office sets a rate for the mahr. Can we give an amount above that rate?

Generally yes, because Islamic law sets no upper limit on the mahr; how it is recorded, however, is governed by state rules. The mahr is the bride's right. Many couples state the mahr at the state rate, then give additional gifts such as gold or money. State clearly what counts as the mahr so there is no confusion later, especially in the event of divorce or death. But do not raise the mahr so high that it burdens the groom or becomes something to show off. Ask the marriage officer in your state how the mahr and additional gifts are recorded on the marriage form. See mahr (dower).