USB sticks are traditionally formatted with FAT 32, because this file system is implemented by almost every operating system and device. Unfortunately, it sucks, as it cannot use more than 2 TiB, store files larger than 2 GiB or store symbolic links for instance. In a word, it is an obsolete and deficient file system.
Good news: someone addressed that problem. Bad new: that someone is Microsoft. So as you could expect, exFAT, the extended FAT, is a stinking proprietary, secret and patented file system. There are free implementations of that shit, but it is safer to stay away from it.
UDF to the rescue!
Good news: there is one file system that is implemented almost everywhere as well, and which does not suffer from such limitations. UDF, the Universal Disk Format, is an ISO standard originally designed for DVDs, but it is perfectly usable for USB sticks. It also supports POSIX permissions, with one killer feature for removable media: a file can belong to no specific person or group.
So, to use it, assuming you USB stick is
$ dd if=/dev/zero of=/dev/sdc bs=1M count=1 $ mkudffs -b 512 --media-type=hd /dev/sdc
The initial dd is to erase the existing partition table or file system information, to prevent you USB stick from being detected as a FAT after it has been formatted with UDF.
-b 512 is to force a file system
block size equal to the USB stick's physical block size, as required by the UDF
specification. Adapt it if you have the luck of having a USB stick with an more
appropriate block size.
After that, your USB stick will be usable for reading and writing with GNU/Linux and the other free operating systems of course, but also with current versions of Windows (read-only with the outdated version XP) and with MacOS.