In recent years, there has been a significant growth in the research area on polygonal methods for solving partial differential equations. One of them, the Virtual Element Method (VEM), gained more and more attention because it allows for elements of arbitrary and highly distorted shapes and varying numbers of edges. However, in most cases, the VEM requires stabilization to cure the rank deficiency of the stiffness matrix. Unfortunately, in some specific cases, the choice of stabilization can affect the results. Therefore, there is a strong interest to develop self-stabilized virtual elements that retain the advantages of the VEM while avoiding the need for stabilization. This work, exploiting a Hu-Washizu framework, presents an alternative formulation of the VEM [1], particularly suited for the development of self-stabilized elements. Two different approaches will be proposed. In the first one, the strain field is enhanced and additional degrees of freedom are introduced. This technique turns out to be equivalent to the approach presented in [2]. A second approach, originally introduced in [3] for potential problems and here extended to elasticity, enlarges again the strain field but without introducing additional degrees of freedom. This is possible thanks to a suitable projection of the displacements virtual shape functions. We present some numerical results demonstrating that these elements can achieve the correct order of accuracy and, in many cases, exhibit higher accuracy than the standard VEM, as well as locking-free behaviour in the nearly incompressible limit.