You can!
   [1]https://modulatum.wordpress.com/.../creating-a-map-in.../

   Use quake3 engine (open source) but instead of getting quake,
   use OpenArena and follow these instructions.

   They're sorta complicated but if you go step by step by step,
   you'll be able to make your own 1st person shooter using the
   same engine used for Call of Duty and all of them for no money
   and once you figure out how to add textures properly, you CAN
   have your apocalyptic map of your dreams!

   [I'm not going to make it _for you_; I just wanted to find "the
   steps" in how to make a quake3 engine style game like the big
   boys do, but for free. I tried it. I made a blank concrete room
   with me, another guy and a rifle. Test proven now I'm happy -
   and all I did is follow these instructions. What's awesome about
   this, is: If you do it, you can release it to XBox,*Playstation,
   Wii U, Steam, the web and Smartphones... so long as you pay
   appropriate licensing fees and follow each of their specific
   rules.

   There. Your path to being a millionaire. And if gamedev isn't
   your thing, point an out-of-work computer geek with ALL THE TIME
   IN THE WORLD on his/her hands to that site and challenge them to
   make you that game. Give them appropriate royalty fees. Done
   smile emoticon

References

   Visible links
   1. https://modulatum.wordpress.com/2008/07/26/creating-a-map-in-gtkradiant-for-openarena/