Yes you can! In this case you will need to create your own cause. It is a simple 4 step process to create your cause and your online campaign. All funds will be sent to your nominated account and you will take care of getting the funds to your recipient.
See what others have done here http://www.mycause.com.au/fundraising-ideas-volunteering
Create your campaign here http://www.mycause.com.au/personal-cause
Donations to a personal causes are not tax deductible.
You can also raise money for a charity at the same time. This split fundraising is perfect where you have to raise money for your trip and a charity. We take care of the appropriate receipting for all parts of the transaction. For example a donor may wish to donate $100 toward your trip and $50 to the charity. They will make a single transaction of $150 but receive 2 receipts. Of course they can leave you a message with their donation too. So make it easy on yourself and your donors - one page for all your fundraising requirements. Go to http://www.mycause.com.au/personal-cause. Look out for the field "Add a charity?"