Because itchy skin is such a big part of the eczema suffering, when that symptom (pruritus or itch) doesn't manifest we tend to assume or think that it must not be eczema or dermatitis. Dry, flaky skin is a characteristic of eczema and could be what your skin is manifesting. There are many different forms of eczema or dermatitis. Psoriasis is another skin disorder that manifests in the form of patches of "raised red skin covered by a flaky white buildup". In the case of eczema or psoriasis, the skin is manifesting an immune problem; and in both eczema and psoriasis, a trigger instigates the breakout.
In your case, many factors could be contributing to the breakout. Because both eczema and Psoriasis relay the message of an immune problem, internal handling of the skin disorder should always be addressed in the process of healing. It is because of the internal condition that the external instigating factor (allergen/irritant) is able to cause havoc to the skin. In addition to the climate drying out your skin, it seems that your soap or cleanser could also be playing a major role in drying out the skin. Because the skin on the face is more sensitive than that of the rest of the body, you may need to use an alternative cleansing agent for the face. Another drying effect to the skin could be the condition of the water (whether or not it is chemically treated). All of these factors (internal and external) together are most probably (in concert) contributing to your dry skin manifestation.
Should you go to a physician to get a specific diagnosis for either eczema or psoriasis, PLEASE stay away from steroid or symptom suppressing medications that mask the problem, don't address the core root of the problem; and as a result, cause more damage to the body and to the skin. You want to focus your attention on restoring the health to your immune system. Water, fruits and vegetables are big players in restoring immune health. Ironically, this most simple form of restoring your bodies good health is rarely (if ever) mentioned by physicians to their patients. In the case of Psoriasis, skin cells pile up on the surface of the body faster than normal causing the excessive dry, flaky skin. The normal period of skin cell maturing and shedding is 28 days; however, in Psoriasis patients the skin cells move more rapidly up to the surface in about 3-6 days. The immune system is sending the "faulty signal" that's producing the rapid response; thus the need to address the internal in the healing process regarding the external skin. You can restore the health to the immune system and change the rapid response by simply implementing good nutrition. The reason that I stress this point is due to the fact that I know most doctors don't stress it; and not only that, act as if that alternative (eating healthy) would bear no effect in changing the skin condition. There is a posting right underneath yours that bears witness to the fact of how a diet change can heal disease. I could even tell you more regarding this (due to the evidence in my own family); however, that would take more time to write than is available to me right now. I've written of my family's success in treating eczema numerous times and you can find out more on some of my previous postings.
To address the needs of your skin externally, search out natural products that will work with your bodies immune system to restore your skin health. If you need direction in that area, then post a quick message and I'll post a list of the types of vitamins/herbs/oils that you may want to look for in your search.