<p>This paper explores a broader class of piecewise linear functions, extending their applicability beyond conventional domains. Piecewise linear functions are typically defined on closed convex domains, but this work introduces a more versatile set of maps known as SW(E m ,T). These maps are linear only on selected subsets of vectors and components, making them suitable for a wider range of applications.<br>The paper establishes an exponential function, F, which maps linear spaces to the set SW(E m ,T). It rigorously proves the uniqueness and existence of a universal element, denoted as *, within this framework. Furthermore, the paper introduces r-subset wise linear skew symmetric maps denoted as Φ = ∑ λμνϕ, demonstrating that they can be fully characterized by their values for λμν and a basis of E.<br>The concept of an r-determinant function is introduced, defined as an r-subset wise linear skew symmetric map Φ: E m → Γ, with Γ being an arbitrary field of characteristic 0. The paper delves into various properties of r-determinant maps, shedding light on their characteristics and utility.<br>Additionally, the paper explores the adjoint of a linear map ψ ∈ L(E,F), where E and F represent linear spaces. It also discusses the development of an r-determinant function using r-cofactors.<br>Furthermore, this work defines extensions of differential forms through r-subset wise skew symmetric maps, paving the way for generalized differential forms. The paper investigates the basis and spaces of these generalized differential forms.</p>