The basic spinor representations of the groups SO(2n) and \(SO(2n+1)\) are of dimension \(2^ n\). The authors consider the unitary group \(U(2^ n)\) and the restriction \(U(2^ n)\downarrow SO(k)\), \(k=2n\) or \(2n+1\). This gives the possibility to apply representation theory for \(U(2^ n)\). The infinitesimal operators of spinor representations of SO(k) are realised in terms of those for \(U(2^ n)\). The weight raising and weight lowering operators for these representations can be constructed. Of course, the explicit construction for general representations is very complicated. The procedure is illustrated for the two-spinor (vector) representations of SO(5) and SO(8), i.e. for the representations with highest weights (1,0) and (1,0,0,0).