I had installed a little box that feeds the signal back up a cable and then to any tv in the house. Works perfect with both my TBox (ie can have either show on any tv so you can flick between 2 things on sky). So yeah it can be done but mine is just called a backfeed.
You can also get wired or wireless remote extenders so you can change channels in your room. Also, does the box have more than one video/audio out? You don't really want to end up having to run your TV in your lounge just so you can watch the TV in your room.
Get a wireless sender with remote extender. Quality can be better than an RF feed around the house. Plus, you'd be able to use the remote and change channels from the other room. Get a 5.8ghz one if you have wi fi in your house to avoid interference.