Washing up liquid also has glycerine in it to give "sparkle" to the dishes. This sticks to optical coatings and damages them.
Use cheap shampoo (as it has almost no additives bar common salt and a few minor others). It is excellent for cleaning glasses (wash with warm water and dry immediately with a clean cloth or toilet paper), and I use it on my tablet and mobile's screen too.
But honestly, in this weather, unless you can point your breath in another direction, even detergent probably won't do it. As a stopgap, try running your (cleaned) glasses under a hot tap then drying them just before using the mask. It'll work for a while - if warmer than the surroundings, no condensation. And it's cheap, but it's a fuss and you risk burning the bridge of your nose!