If you want to use steel wool, use the finer 0000 steel wool, and wrap it around a bore brush. The bristles will hold the wood to the brush.
YOu can also use products like JB Bore Cleaner.
The problem usually is that someone heated the barrel too hot when last cleaning it- ususally using boiling water in the barrel- which is not necessary . Water is the solvent. Hot water will speed the process of cleaning, but skin temperature, or " tepid " water will work as well, and won't burn your hands, OR produce flash rust in the barrel.
Once the water is dried from the barrel, wipe the barrel with a cleaning patch coated with a good oil or lube, like T/C's Wonderlube. Or use a product called Ballistol. Both will protect the bore during storage.
Remember to clean the stuff out of the barrel before you take the gun shooting again. Alcohol on cleaning patches, and poured down the barrel to clean the powder chamber, and the flash channel at the back of the barrel do a good job of getting all the grease and oil out of the gun.
Then, put a thin-- I DO MEAN THIN- coat of oil on a new cleaning patch and run it down the barrel to protect the lands and grooves during transporation from home to range or field. A dry cleaning patch will remove most of that fine coat of oil and make the gun ready to load with powder.