Defining an abstract Lie group Whenever the group $SU(N)$ is presented, it is usually given as
\begin{equation}
SU(N)=\{U\in GL(N,\mathbb{C})| UU^\dagger=1, \text{det}(U)=1\}.
\end{equation}
However, it seems to me that this is really the definition of the fundamental representation of $SU(N)$. So if I were to ask myself how to give a definition of the abstract $SU(N)$ group, my tendency would be to instead start from the algebra, and say that an element of the abstract $SU(N)$ group is obtained through the exponential map from an arbitrary linear combination of the Lie algebra elements. The Lie algebra is abstractly defined by its commutation relation, and so everything is nice and well behaved.
This cannot be done, however, if the group is not connected, for example, as the exponential map allows only to reach the elements of the component that contains the identity element. So in this case, to define an abstract Lie group, I would have to know its Lie algebra and the center of the group or its topology.
My question, in short, is: how do we define abstract Lie groups in practical cases, and in a way that does not hold for simple groups only? I hope the question makes sense.
 A: You're right. Different Lie groups can have the same Lie algebra, so the Lie algebra is not enough.
You asked how to define abstract Lie groups in practical cases. Different people will have different opinions of what practical means. People in engineering disciples will think a matrix parameterization is the most practical thing out there. There is also the Adjoint representation of a Lie group, which might be what you're looking for.
A: One thing I would note is that a representation is not the same as a matrix representation. We don't need to choose a basis for a representation. So in your example we could say a little more abstractly that $SU(n)$ is the set of linear transformations preserving a (complex) inner product on $\mathbb{C}^n$
It is very normal to define a Lie group by one of its representations in this way and indeed in maths we usually call these the "defining representations". Note physicists call these the fundamental representations but in maths that refers to a broader group of representations.
If we want to get a more abstract definition we can use the Lie algebra but we have to be careful. There isn't one exponential map for a given Lie algebra. For every Lie group $G$ with Lie algebra $\mathfrak{g}$ there is an exponential map $\mathfrak{g}\to G$. We have to define $G$ already to get this map so we cannot use this as a definition of $G$.
As Torsten points out however we can find all the connected Lie groups with a given Lie algebra in a more abstract way. There is a unique (up to isomorphism) connected, simply-connected Lie group $\tilde{G}$ with Lie algebra $\mathfrak{g}$. All other connected Lie groups with Lie algebra $\mathfrak{g}$ are quotients of $\tilde{G}$ by discrete subgroups of its centre. If $\mathfrak{g}$ is semisimple this centre itself is discrete so we can classify the groups easily.
