How to enable Node editing in full mode?
Yes there is a module which can enable this functionality - Editablefields.

Module page says itself --


  This module allows CCK fields to be edited on a node's display (e.g.
  at node/123), not just on the node edit pages (e.g. node/123/edit). It
  also works within views etc. Anywhere a 'formatter' can be selected,
  you can select editable (or click to edit).


I have used this module for edit fields at views not at node, but I am sure you can achieve your requirement.