Can an already married couple turn their marriage into a civil union?

My wife and I have been happily married for three years now, but we aren't religious in any way and we don't want to be ociated with the term "marriage." Marriage has become synonymous with the bigoted "one man, one woman" Christian beliefs. Since we aren't Christians, and we want to support the right of all people to be equal under the law, we'd like to have a civil union instead. Is there anyway to turn a marriage into a civil union?