That's exactly what I did. We cut a square out over the drain and kept the square. I can plug it when the farrier comes (does three at a time, so one in the wash stall).
I originally had ring mats and hated them. They collected dirt in the holes and I had to pull them out every couple of weeks and clean. Now I just pour some bleach under the mats every month or so in the summer.
The wash stall actually drains better with the mats. Love it!
We have heavy rubber mats that are in a mesh pattern so they drain and provide traction. Dirt and debris still tend to accumulate under them but they drain well and provide traction. They are about 4' x 4' so they are easy for one person to move and clean under as well.
That's what we did, and it's worked out well. Only thing that we did wrong was ours fit so snugly they're a pain to pull up to put bleach underneath every so often. So give yourself a little wiggle room.
"Our greatest glory is not in never falling, but in rising every time we fall." - Confucious