Yes, one can maintain a reef tank without a skimmer.
Using Ecosystem for example, the export of nutrients is utilized by algae turf. I have seen many reef tanks using this method without the installation of any skimmer. However, after two to three years you will need to replace your system's mud to prevent substrate over load.
This system is simple. Set up an outside filter that could be hang- on the back of your tank. Usually in a capacity that is about 1/4 of your total tank size. I think the other reefer on this board named Andy built a hang on back refugium. You can do a search on his posts. Then plant the filter with fine sand or mud. Preferably aragonite sand, special grade mud or rubble live rock that are high in carbonate and iron. Then you can plant the algae. Supply the algae with strong light and have it 24/7. You should trim the alge as the filter starts to functioning on a regular basis.
I personally prefer the skimmer as I believe this is the best way to control undesirable nutrient build up. HTH!