You are only as strong as your weakest link and in this case, and I am similar, it is your triceps.

What I would suggest is making sure you have a good tricep routine as well and do it on a different day to chest (do it after chest day to )

Two really good tricep exercises are

Close grip bench press - Hands 3-5 inches apart on the centre of the bar (watch your wrists)

and

Skull crushers - lying on a bench with an EZ bar, upper arms/elbows pointing to the ceiling and then lowering the bar to your forehead and press back up. Start light and work your way up. too heavy too quickly and you'll hurt your elbows.

These two can add real strength to your tri's and help with the all important lockout phase which is what it sounds like you are struggling with.

I personalluy have never got on with flat benching as my shoulders have always taken over too much. As was stated above, using dumbells really helps, along with doing dumbell flyes where you have your palms facing each other and lower the bells in an arch out to your sides.

Also to help your chest and your triceps, do wide grip dips and progress to hanging weight from your waist. The gym should have a belt with a chain on so you can hang weights. This helps the outside of your chest as well as strengthen your triceps.

Also - leave your ego at the door. If you can't lock out fully with a weight for at least 5 reps, then it is too heavy. Drop it back down and build up slowly so you can do it properly. You may find you only need to take of 5 kg.