You definitely can. However, you will be reinventing the wheel every time you want to develop a new game.
A game engine is a set of rendering and mathematical operations that simplifies the complexity of game development.
In layman’s term, it provides a set of tools that you can re-use every time you need to develop a new game.
For example, to build a house you need, at a minimum, the following tools: hammer, nail, saw, wood and a blueprint.
No matter how many houses you build, the only items that changes are the wood, nails and the blueprint. There is no need for you to manufacture your own hammer and saw. You simply buy them once and reuse them every time you build a new house.
A game engine provides you the algorithms and components, i.e., hammer and saw, to build a game. You simply provide the characters and the gameplay. i.e., nails, wood and blueprints.
So what exactly does a game engine provide?
A game engine provides you with the following components:
Math Engine: Responsible for all Linear Algebra computations, such as Space Transformations.
Rendering Engine: Responsible for interacting with the GPU and manages all Shaders (GPU programs) accountable for rendering 3D models and various graphics effects, such as shadows, lighting and particle systems.
Physics Engine: Responsible for solving the equation of motion
Collision-Detection System: Responsible for detecting collision among Convex objects.
You can definitely develop a game without a game engine, but you will need to implement the components mentioned above. Moreover, developing these components require a different set of expertise than what is required in game development.