No you do NOT need a special cleaner for that! 

Have you looked at the upper rim of your washing machine drum? Mine was pretty gross. I don't even know what prompted me to look there, but I caught a glimpse of it and immediately felt the need to clean it. How does all that gook get there? My washing machine is just over a year old and has never been clean. My previous washing machine was just over 7 years old and I never even thought to look at the drum to check for gunk. Yikes!

Isn't that gross? First I took a wet washcloth and put a squirt of laundry soap on it and scrubbed it. Then I thought I was going to have break out an old toothbrush or the cotton swabs. I really didn't want to get that crazy with it so I grabbed some scissors to jam my washcloth into the knicks, crannies and ridges. It did the trick!
I then took a dry washcloth and poured some vinegar on it and wiped it all down until it was all sparkly.

Now my washing machine drum is nice a clean! Try it :)