It may surprise you to learn that the professional fighters here in Thailand don't stretch at all! At least not where I'm training.
However, I've found a general all over body stretching routine works the best. It takes me about 10 minutes to stretch everything.
To get your elbows on the mat during the stretch you describe isn't as easy as it looks and of course you can only stretch until a certain point before it starts to hurt. Try and improve gradually each day, stretching just that little bit more. Make sure you stretch all your muscles, not just the problem ones.