It is easy to construct any operator (in continuous variables) using the set of operators {|ℓ⟩⟨m|},
Actually, this boils down to a single example which would be sufficient. Can we find coefficients αk,ℓ such that |0⟩⟨0|=∑k,ℓαk,ℓak(a†)ℓ.
Answer
@Accidental reminds you this is a theorem. To actually see it in your terms, use the infinite matrix representation of a,a† of Messiah's classic QM, v 1, ChXII, § 5. Specifically, your vacuum projection operator has a 1 in the 1,1 entry and zeros everywhere else.
The operator you chose is freaky to represent, but, purely formally, the diagonal operator for N≡a†a, |0⟩⟨0|=(1+N)(1−N)2−N23−N34−N4...
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