After changing all the strings and tuning the guitar I went back to the course. After some chords I came to the F-chord. Set my fingers, strumm and....a clear sound. A beginners luck, so I tried again. It kept giving me a clear sound. Now I do not know what the original strings were on the guitar, but my E-string now has a thickness of 0.1 inch. So, I guess that made the big difference. I can not do a full barred F-chord, but at least without using the lower E I can get that chord. Now all that is left is to build up the so called muscle memory in order to be more quick in changing chords.
So, for all that have such an issue with this chord I suggest to have a look at the thickness of your strings. Instead of attacking this chords for weeks in a row, you could save yourself some time and frustration by using a different type of string.
Regards,
Jan