What do you mean by "alias"? A symbolic link to /etc/skel/.bashrc? Doesn't adduser automatically copy in a fresh .bashrc for you?

If you are editing .bashrc and it's a symbolic link, you need write permission to the target of the symbolic link. But instead, you should have a separate file owned by the user, usually a copy of /etc/skel/.bashrc

If you are editing a file owned by a different user, you need to be root, or the file's permissions need to be relaxed enough to allow you to overwrite the file (and, the way editors often work, probably write access to the directory). This is of course not really a good idea in any event.

.bashrc doesn't need execute permission, it's sourced (not executed) by bash.