1999 Honda Accord Were is the oil going?

Hello, my name is dan. Been having problems with oil loss in the past 8 months. After about 1,000 miles most of my oil is gone. The check engine light comes on, but no oil low light(it works). At first it cost me on repairs that had nothing to do with the low oil pressure code. Now I just keep the tank full and I'm ok. No signs of leak, no signs of burning, car performs great, no smoke on acceleration or going up a high grade hill. Could this car be burning oil this fast and still drive as well as it does? When the oil does get low the check engine light comes on and on freeway acceleration the car cuts out. Any suggestions?