The best way to test this is by keeping a dream journal by your bed. The key of this trick is every time you have a dream you will wake up and write it down. Now since you say you do not remember your dreams it will take some time to train your brain to do so with this journal. Kind of like hitting two birds with one stone kind of trick.
Let me recommend you few things that may help you with your problem, in addition to what Nash said.
Before going to sleep, what i recommend is to ffall asleep to the mantra "I will remember my dreams". It may sound really silly for few folks who get to hear this mantra, but I do this everynight for a week before asking for questions to be answered through my dreams to make sure I will remember any answers given.
Eventually you will begin remembering bits and pieces. When you wake, don't get up right away. Try not to move at all. People who don't have good dream recall are usually ones who jump out of bed immediately. Think back about that night and go over any dreams or fragments you remember. Once you've consciously gone over anything that was recalled that night, sit up SLOWLY, continuing to concentrate on what you remember. Have a notebook (it's basically called a 'dream journal' as Nash explained to you) and pen ready at the side of your bed. Write down every detail that you remember. This should be also done if you wake up in the middle of the night too.
Do this everynight for two weeks and you'll notice a big difference. Do it for a month and you'll begin remembering full dreams, sometimes even multiple ones depending on how pertinacious you are with this method.