Let $M,N$ be two smooth connected real $n$-dimensional manifolds. Suppose they are both orientable for simplicity. Let $i:D_1\hookrightarrow M$ and $j:D_2\hookrightarrow N$ two embedded disks so that $i$ preserves the orientation and $j$ reverses the orientation. Consider their connected sum given by \begin{equation*}M\#N=(M\setminus\overset{\circ}D_1)\sqcup_\phi(N\setminus\overset{\circ}D_2) \end{equation*}where $\phi:\partial D_1\rightarrow\partial D_2$ is a diffeomorphism glueing the boundary of the two disks. The hypothesis on $j$ to reverse the orientation is given in order to ensure that $M\#N$ is orientable.
At this point I would like to know if there is a canonical approach to understand how the tangent space at a point of the connected sum looks like and if this can help us to have global information on the tangent bundle. Why Am I asking that?
I want to understand some characteristic classes associated to the connected sum. For example we can consider the total Stiefel-Whitney class $w(M\#N)\in H^*(M\#N, \mathbb{Z}_2)$ or the Euler class $e(M\#N)\in H^n(M\#N,\mathbb{Z})$. I was thinking about the case when the tangent bundle splits as a direct sum (product of manifold $M\times N$) and so by using Whitney product formula for the Stiefel-Whitney class we can recover $w(M\times N)$ from $w(M)$ and $w(N)$. In particular this seems to me no longer true for the connected sum even in the simplest case of connected compact and orientable surface.
Question
Is there a canonical approach to understand how $T(M\#N)$ looks like and to get information on its characteristic classes? Or better, is there a way to understand how for example $w(M\# N)$ looks like by using only the axioms and using only the construction of the connected sum?