Food allergies are typically presented in dogs with the itching of the skin. However, other signs of food allergies can vary depending on the severity between animals.
The best way to determine if your dog is suffering from a food allergy is to see your veterinarian. If your veterinarian rules out infection or other diseases, he or she will then do an “elimination.” This is a diet that is nutritionally complete, yet it does not contain any of the ingredients that the dog had previously been exposed to. This is usually done for a period of a few months. If symptoms disappear, then a food allergy is most likely the main suspect.
Next, previous food ingredients will be fed back to the dog to try to pinpoint which exact ingredient is causing the allergy. Some of the most common food allergens seen in dogs tend to be beef, chicken, eggs, corn, soy, and milk. Many pet owners are reluctant to harm their pets, but this strategy directly pinpoints the problem food so your animal can eat complete meals again. Some pet foods that are known to be "hypoallergenic" include brands such as Blue Buffalo®, Natural Balance®, Rachel Ray’s Nutrish® dog food, and similar products. You can find these brands at your local pet store or order them online.
Some pet owners will elect to home cook a diet for their animal, but for owners who are less kitchen-savvy, your dog can still eat happy and healthy from a hypoallergenic dog food.
If allergens are not the problem, some skin disorders can also explain flaking skin and hair loss. Sarcoptic mange is caused by mites burrowing into the skin. According to the article Ectoparasites: Sarcoptic and Demodectic Mange Mites, "Dogs with sarcoptic mange will suffer from intense itchiness and lesions, and in advanced stages can undergo hair loss and severe debilitation. Because the lesions can become extremely itchy, they are often the cause of self-mutilation; if left untreated, they can become bloody and infected. The areas most commonly affected are the margins of the ears, the knees, and feet, although lesions may also be observed on the flanks and stomach.”
A visit with your veterinarian will likely determine the exact cause of your dog's flaking skin and hair loss, as well as appropriate treatments.
For more detailed information, see this article from eXtension: Ectoparasites: Sarcoptic and Demodectic Mange Mites.