Is there a Tight frame like this? Let $V$ a finite dimensional vector space of dimension $n$. I want to answer the following question: 
Is there a tight frame $$\Phi=\{e_1,\cdots,e_n,\cdots,e_m\}$$ such that if you consider any subset of $n$ elements then is still a frame?. Not necessarily tight. 
PD: I think the answer is yes! How ever, i can't prove it. I don't know the method to approach it. Any help is welcome. All that i know is that a tight frame is a frame such that there exist a positive constant $A$ such that $$A||x||^2=\sum_1^m|<x,e_k>|^2, \forall x\in V$$

It sounds like you are describing a Tight Frame with General Position
It is trivial to construct from a frame of random orthobases:
octave:1> A=[orth(randn(3));orth(randn(3));orth(randn(3)) ];
octave:2> svds(A) % tight frames have uniform singular values 
ans =
   1.7321   1.7321   1.7321

octave:3> ix=randperm(9); ix=ix(1:3);  % any subset of rows ...
octave:4> cond(A(ix,:))   % will also be a frame (cond<inf)
ans =    
   3.24748

