See my edit below
Try this :
RewriteEngine On
RewriteRule ^(.*)$ ?path=$1 [QSA]
The first part of the RewriteRule will capture the full path without the query string (GET parameters). The second part of the RewriteRule will rewrite with a GET parameter named "path" containing the full path we captured before. And then, using the [QSA] flag (which should be default behaviour of Apache), we tell Apache to keep the original GET parameters and append then to the new query string.
EDIT :
There is a major problem with this answer, a redirection loop. Here a better solution :
RewriteEngine On
RewriteCond %{QUERY_STRING} !path=
RewriteRule ^(.*)$ ?path=$1 [QSA,L]