It's fairly easy to do this. You can use a widget anywhere plugin then move the search code to it. Done!
Install Widget Anywhere. (If you don't have it already. I've found it extremely useful in many ways.)
Go to admin > plugins > widget anywhere > add new widget
Set it to main area - top.
Copy your search bar code by inspecting the search element in your browser and copying the entire <div> that contains it. For instance, the search for this site is:
<form method="GET" action="../search">
<input type="text" name="q" value="" class="qa-search-field">
<input type="submit" value="Search" class="qa-search-button">
Yours will likely be the same, unless you changed some stuff. That's it! You can specify where you'd like this to show up in Widget Anywhere. This will populate the search bar and submit button.