Proposition 11
Tag 05NO, permanent. Cite it as [Bourbaki, Tag 05NO].
Algebra, chapter II, § 2. Modules of linear mappings. Duality
The dual of a free module with a basis of n elements is a free module with a basis of n elements.
Note that the dual of a free module with an infinite basis is not necessarily a free module (VII, § 3, Exercise 10).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.