Can I use touch up paint to fix this big scratch?

Yes but it will show. If you want it to look good then have a paint shop repaint the whole bumper.

Oh man, they charge $400 for that. Is there any way for it to not show?

If you carefully paint it and then color sand it with really fine sand paper and then buff it out it might look good if you are 20 feet away.

Read this on color sanding ----- http://www.meguiarsonline.com/forums/showthread.php?29980- What-grit-papers-for-wet-sanding#kYscUoGkjIFKaMjv.97

Look at the color code on the cars detail sticker either at door jam or under hood and buy a can of that color (spray can) and tape a area (rectangle) just larger then the scratch and lightly sand inside or the taped area with like 200-400 grit and put paper down to protect the rest of car and spray it, if the can of paint does not have clear coat in it then also buy clear and hit it when it dries with clear, be sure to put down enough color and lightly sand between coats using finer and finer grit as you go. you will need to sand down to the same depth the scratch is to make it so it is not seen. 4-7 coats color and 2-3 coats clear

Then color sand and/or polish the whole bumper.

Thank you, everyone. Can you give me a list of things I need to buy to do this?

use the 200-400 grit to start out then in between coats move to higher grit up to about 2000 grit

may have to find the color at a auto paint store to get it in a spray can , some tap like painters tap, the sand paper in the different grits, paper or plastic to cover the car. some wet sanding grit 2500-3000 grit or higher and used in conjunction with water and/or some buffing compounds if using a buffer just go very lightly here, look at some you-tube videos before starting out and when doing the whole bumper to blend it in you are just taking enough off to match the new paint if it does not already match up because of age or sun fade ect.

If the scratch has a deep spot into plastic the some filler too needs to be added to that list.

Plastic filler if needed.

Thank you to you both. I'll upload a pic when I'm done.

That is a flexible surface that has been flexed to where the paint is loose. More paint will likely come off. You will not be able to fix it to where it doesn't show without professional help.