This is most likely a permission problem. If you do

sudo mkdir test
sudo chown your_user_id test

you should be able to create your own stuff under test.

You can also mount the partition under your own id:

mount  -o uid=ownerID /dev/sda1 /mountpoint

This is most likely a permission problem. If you do

sudo mkdir test
sudo chown your_user_id test

you should be able to create your own stuff under test.

If you chown and/or chmod the top of the file system when it is mounted, it should be accessible for the right users.

One word of caution: do not make everything world-writable. If you have multiple users, give them their own directory (like you did with test)