I'm sorry to hear about your dog‘s eye. If attempting to treat at home you want to be very careful first of all that the eye continually looks better through treatment, and no signs of rubbing at it, reddening worse, swelling worse or staying the same, or cloudiness appear in the eye.
You should rinse the eye with sterile saline, like that found in regular contact lens solution, and do not let the tip of the container touch the eye. Make sure it is not the disinfecting/storing type of contact lens solution, just the normal saline/ NaCl. You can wipe around the eye with the saline soaked on a soft cotton ball or gauze, but don't use anything harsh like paper towels. Try your best to gently but firmly open the eyes while rinsing; you may need a second set of hands.
You can also find an ophthalmic antibiotic ointment at a pharmacy: like terramycin, triple antibiotic (neo/poly/bac), etc. Ask the sales people if you need help, but it does need to be safe for the eye, not a topical one for skin. After rinsing the eyes, apply the ointment 3 times a day (every 8 hrs) for 10 to 14 days. It needs to go in the eye (not the tip, just the ointment)- do your best to open each eye and squirt just a 1/4" strip onto the outer eyeball, it will spread on its own.
This may be conjunctivitis, but could also be a scratch or ulcer forming and if not starting to clear up in a couple of days, you'll need a vet to check it. It is also very possible some debris or dust settled in your dog's eye, causing this problem. Going on for too long can cause more permanent damage, so don't wait too long for a vet appointment for a true diagnosis!
Best of luck, and let me know if I can help further.