Bit difficult to do so. In case of Raspbian and Archlinux you have only a user that is an ordinary user. Ordinary users have limited access to system files and folders (sometimes they can only read it, sometimes they can't even do that). You use sudo to elevate that user to root privileges because that's what you need to access and write system files/folders.
Applications such as Forklift need to support it but from what I know none do. You could change the permissions to allow ordinary users to do that as well but this can break the system and poses security risks (the system is set up like this because of security). The only other option is to use Terminal and ssh to log into the Pi and work locally on it via the ssh tunnel. You can then use sudo to do whatever you need to do that requires root privileges.
However, this does not explain why you could read and write before. If you use an ordinary sd card it's best to check if you haven't turned the lock switch to the lock position. If you did then the sd card will be in read only mode and not allow you to write files to it, not even when being root. In some cases the switch gets flipped upon insertion of the sd card. Might have happened to you. If that isn't the case you should run a filesystem/integrity check to see if the sd card is broken.