<p>A new paradigm for deriving 2D Hermite basis based solely on 1D Lagrange polynomials is presented in this paper. This approach significantly simplifies the development of a novel weak-form quadrature element method (WQEM) for 2D fourth-order systems present in the nano-mechanics field as well as thin plate problems. The new formulation is computed based on regular 1D Lagrange-based DQM while also giving access to the analytical expression of the new 2D Hermite basis. This is achieved through the development of purpose build transfer matrices between a number of 2D polynomial bases. A key advantage of the proposed WQEM formulation is the use of a non-tensor product basis and the development of high-order 2D Hermite polynomials without second derivatives at the corner points. This solves the mismatch between the physical degrees of freedom and those provided by the WQEM formulation generally found in classical implementations. Furthermore, using the analytical expression of this new basis and a set of matrix transformations, full integration was achieved for the proposed WQEM element. The free vibration of skew thin plates with various skew angles and different combinations of boundary conditions is used to assess the convergence and the accuracy of the proposed formulation. Convergence studies demonstrated that, despite its relative simplicity, the proposed method achieves convergence rates comparable to, or faster than, other existing formulations while maintaining similar accuracy. The method’s performance and accuracy were evaluated against data from the literature and finite element method simulations, revealing that the proposed WQEM yields precise frequency results even for significantly skewed angles. Application for the present approach to generate DQM matrix-based high-order formulations is discussed in Part II.</p>