Sorry, but the DOT rule is that a carrier must offer either a fee-free refund or hold. AA offers the hold. Thus, no refund.

It's a toll-free call, so it won't hurt you to explain your error and ask whether AA might issue a refund, but don't hold your breath.

However, presuming that you purchased a standard non-refundable ticket, it should retain its value less a $200 penalty (typically) which your wife may use as a credit against other travel for a year.