Vacuum or sweep up any loose debris on the concrete floor.
Use a utility knife to cut an "X" shape into the side of a new tennis ball. Do no use an old tennis ball; you need the ball to be clean and new. Insert the bottom end of a broom handle into the "X" you just cut. Place the tennis ball on the floor and press it down on any scuff marks, in a mopping type motion. The tennis ball will act like a squeegee, and will remove some of the scuff marks.
Fill a bucket with warm water. Dip a stiff-bristled cleaning brush into the water. Scrub the scuff marks with the brush. Refrain from using a brush with metal bristles, as this can scratch and damage the concrete floor.
Pour out the water from the bucket. Add 1 tbsp. of liquid dish soap to the bottom of the bucket. Fill the bucket with 1 gallon of warm water. Pour 1 to 2 tbsp. of ammonia into the soapy water. Stir the water with the brush, then scrub any remaining scuff marks with the water mixture. Wipe away the excess water with a sponge.